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Finding Zeros By Completing The Square Common Core Algebra 1 Homework Answers
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Kum Alcini
Dec 8, 2023, 8:23:32 PM12/8/23
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Finding Zeros By Completing The Square Common Core Algebra 1 Homework Answers
Finding zeros by completing the square is a technique that can help you solve quadratic equations that are not easily factorable or have irrational roots. In this article, you will learn how to use this technique and how to apply it to common core algebra 1 homework problems.
What is Completing the Square?
Completing the square is a method of transforming a quadratic equation into a perfect square form. A perfect square form is an expression that can be written as the square of a binomial, such as (x+3)^2 (x +3)2 or (x-5)^2 (x 5)2. A perfect square form has the advantage of being easy to solve by taking the square root of both sides.
Finding Zeros By Completing The Square Common Core Algebra 1 Homework Answers
Download Zip https://t.co/L8JpcnnLCO
To complete the square, you need to follow these steps:
Move the constant term to the right side of the equation.
Divide both sides by the coefficient of x^2 x2 , if it is not 1.
Add the square of half of the coefficient of x x to both sides.
Factor the left side as a perfect square.
Solve for x x by taking the square root of both sides and simplifying.
Why is Completing the Square Useful?
Completing the square is useful for several reasons:
It can help you find the zeros or roots of a quadratic equation, which are the values of x x that make the equation equal to zero.
It can help you find the vertex or turning point of a quadratic function, which is the point where the function reaches its maximum or minimum value.
It can help you graph a quadratic function by finding its vertex and axis of symmetry.
It can help you solve quadratic inequalities by finding the intervals where the function is positive or negative.
How to Find Zeros By Completing the Square in Common Core Algebra 1?
In common core algebra 1, you may encounter homework problems that ask you to find zeros by completing the square. Here are some examples and solutions:
Example 1
Solve x^2+6x-7=0 x2 +6x 7 = 0 by completing the square.
Solution:
Move the constant term to the right side: x^2+6x=7 x2 +6x = 7
Divide both sides by 1: x^2+6x=7 x2 +6x = 7 (no change)
Add the square of half of the coefficient of x x to both sides: x^2+6x+9=16 x2 +6x +9 = 16
Factor the left side as a perfect square: (x+3)^2=16 (x +3)2 = 16
Solve for x x by taking the square root of both sides and simplifying: x+3=\\pm\\sqrt16 x +3 = 16
x=-3\\pm4 x = 3 4
x=-7\\text or x=1 x = 7 or x = 1
The zeros are -7 -7 and 1 1 .
Example 2
Solve 3x^2-12x-15=0 3x2 12x 15 = 0 by completing the square.
Solution:
Move the constant term to the right side: 3x^2-12x=15 3x2 12x = 15
Divide both sides by 3: x^2-4x=5 x2 4x = 5
Add the square of half of the coefficient of x x to both sides: x^2-4x+4=9 x2 4x +4 = 9
Factor the left side as a perfect square: (x-2)^2=9 (x 2)2 = 9
Solve for x x by taking the square root of both sides and simplifying: x-2=\\pm\\sqrt9 x 2 = 9
x=2\\pm3 x = 2 3
x=-1\\text or x=5 x = 1 or x = 5
The zeros are -1 -1 and 5 5 .
Example 3
Solve \\frac 1 4x^2-\\frac 1 8x-\\frac 5 8=0 \\dfrac14
How to Find Vertex and Axis of Symmetry By Completing the Square in Common Core Algebra 1?
Completing the square can also help you find the vertex and axis of symmetry of a quadratic function. The vertex is the point where the function reaches its maximum or minimum value, and the axis of symmetry is the vertical line that passes through the vertex and divides the graph into two congruent parts.
To find the vertex and axis of symmetry by completing the square, you need to follow these steps:
Write the quadratic function in standard form: y=ax^2+bx+c y = ax2 +bx +c
Complete the square for the expression ax^2+bx ax2 +bx and factor it as a perfect square.
Write the quadratic function in vertex form: y=a(x-h)^2+k y = a(x h)2 +k , where (h,k) (h, k) is the vertex.
Find the x x -coordinate of the vertex by setting x=h x = h and solving for x x .
Find the y y -coordinate of the vertex by plugging x=h x = h into the function and solving for y y .
Find the equation of the axis of symmetry by writing x=h x = h .
Example 4
Find the vertex and axis of symmetry of the quadratic function y=2x^2-8x+5 y = 2x2 8x +5 by completing the square.
Solution:
Write the quadratic function in standard form: y=2x^2-8x+5 y = 2x2 8x +5 (no change)
Complete the square for the expression 2x^2-8x 2x2 8x and factor it as a perfect square: y=2(x^2-4x+4)-8+5 y = 2(x2 4x +4) 8 +5
y=2(x-2)^2-3 y = 2(x 2)2 3
Write the quadratic function in vertex form: y=2(x-2)^2-3 y = 2(x 2)2 3 , where (h,k)=(2,-3) (h, k) = (2, 3) is the vertex.
Find the x x -coordinate of the vertex by setting x=2 x = 2 and solving for x x : x=2 x = 2 (no change)
Find the y y -coordinate of the vertex by plugging x=2 x = 2 into the function and solving for y y : y=2(2-2)^2-3 y = 2(2 2)2 3
y=-3 y = 3
Find the equation of the axis of symmetry by writing x=2 x = 2 .
The vertex is (2,-3) (2, 3) and the axis of symmetry is x=2 x = 2 .
Example 5
Find the vertex and axis of symmetry of the quadratic function y=-\\frac 1 4x^
How to Solve Quadratic Inequalities By Completing the Square in Common Core Algebra 1?
Completing the square can also help you solve quadratic inequalities, which are expressions that involve a quadratic function and an inequality sign, such as , \\leq \\leq , or \\geq \\geq . A quadratic inequality can have one or two solutions, depending on the sign of the leading coefficient and the direction of the inequality.
To solve quadratic inequalities by completing the square, you need to follow these steps:
Write the quadratic inequality in standard form: ax^2+bx+c0 a x2 +bx +c > 0 , ax^2+bx+c\\leq0 a x2 +bx +c 0 , or ax^2+bx+c\\geq0 a x2 +bx +c 0
Complete the square for the expression ax^2+bx ax2 +bx and factor it as a perfect square.
Write the quadratic inequality in vertex form: a(x-h)^2+k0 a (x h)2 +k > 0 , a(x-h)^2+k\\leq0 a (x h)2 +k 0 , or a(x-h)^2+k\\geq0 a (x h)2 +k 0 , where (h,k) (h, k) is the vertex.
Find the zeros of the quadratic function by setting it equal to zero and solving for x x . These are the boundary points that divide the number line into intervals.
Test each interval by plugging in any value of x x from that interval into the quadratic function and checking the sign of the result. If the result satisfies the inequality, then the interval is part of the solution. If not, then the interval is not part of the solution.
Write the solution as an interval notation or a set notation, using parentheses or brackets depending on whether the boundary points are included or not.
Example 6
Solve x^2-6x+5Solution:
Write the quadratic inequality in standard form: x^2-6x+5
Complete the square for the expression x^2-6x x2 6x and factor it as a perfect square: x^2-6x+9
Write the quadratic inequality in vertex form: (x-3)^2
Find the zeros of the quadratic function by setting it equal to zero and solving for x x : (x-3)^2=4 (x 3)2 = 4
x-3=\\pm\\sqrt4 x 3 = 4
x=3\\pm2 x = 3 2
x=1\\text or x=5 x = 1 or x = 5
Test each interval by plugging in any value of x x from that interval into the quadratic function and checking the sign of the result: Interval Test value Result Solution (-\\infty,1) (-\\infty,1) -1 -1 (-1)^
How to Graph Quadratic Functions By Completing the Square in Common Core Algebra 1?
Completing the square can also help you graph quadratic functions, which are functions that have the form y=ax^2+bx+c y = ax2 +bx +c , where a a is not zero. A quadratic function has a U-shaped curve called a parabola, which can open up or down depending on the sign of a a .
To graph quadratic functions by completing the square, you need to follow these steps:
Write the quadratic function in standard form: y=ax^2+bx+c y = ax2 +bx +c
Complete the square for the expression ax^2+bx ax2 +bx and factor it as a perfect square.
Write the quadratic function in vertex form: y=a(x-h)^2+k y = a(x h)2 +k , where (h,k) (h, k) is the vertex.
Plot the vertex on the coordinate plane as a point.
Find the x x -intercepts by setting y=0 y = 0 and solving for x x . These are the points where the parabola crosses the x x -axis.
Plot the x x -intercepts on the coordinate plane as points.
Find the y y -intercept by setting x=0 x = 0 and solving for y y . This is the point where the parabola crosses the y y -axis.
Plot the y y -intercept on the coordinate plane as a point.
Draw a smooth curve that passes through all the points and opens up or down depending on the sign of a a .
Example 7
Graph the quadratic function y=x^2-4x-5 y = x2 4x 5 by completing the square.
Solution:
Write the quadratic function in standard form: y=x^2-4x-5 y = x2 4x 5 (no change)
Complete the square for the expression x^2-4x x2 4x and factor it as a perfect square: y=x^2-4x+4-9 y = x2 4x +4 9
y=(x-2)^2-9 y = (x 2)2 9
Write the quadratic function in vertex form: y=(x-2)^2-9 y = (x 2)2 9 , where (h,k)=(2,-9) (h, k) = (2, 9) is the vertex.
Plot the vertex on the coordinate plane as a point: (2,-9) (2, 9)
Find the x x -intercepts by setting y=0 y = 0 and solving for x x : 0=(x-2)^
Conclusion
Completing the square is a powerful technique that can help you solve and graph quadratic equations and functions. It can help you find the zeros, vertex, axis of symmetry, and intervals of a quadratic function. It can also help you transform a quadratic function into a perfect square form that is easy to work with. In this article, you learned how to use completing the square in common core algebra 1 homework problems and examples. By following the steps and tips in this article, you can master completing the square and ace your algebra 1 tests.
a8ba361960
Finding zeros by completing the square is a technique that can help you solve quadratic equations that are not easily factorable or have irrational roots. In this article, you will learn how to use this technique and how to apply it to common core algebra 1 homework problems.
What is Completing the Square?
Completing the square is a method of transforming a quadratic equation into a perfect square form. A perfect square form is an expression that can be written as the square of a binomial, such as (x+3)^2 (x +3)2 or (x-5)^2 (x 5)2. A perfect square form has the advantage of being easy to solve by taking the square root of both sides.
Finding Zeros By Completing The Square Common Core Algebra 1 Homework Answers
Download Zip https://t.co/L8JpcnnLCO
To complete the square, you need to follow these steps:
Move the constant term to the right side of the equation.
Divide both sides by the coefficient of x^2 x2 , if it is not 1.
Add the square of half of the coefficient of x x to both sides.
Factor the left side as a perfect square.
Solve for x x by taking the square root of both sides and simplifying.
Why is Completing the Square Useful?
Completing the square is useful for several reasons:
It can help you find the zeros or roots of a quadratic equation, which are the values of x x that make the equation equal to zero.
It can help you find the vertex or turning point of a quadratic function, which is the point where the function reaches its maximum or minimum value.
It can help you graph a quadratic function by finding its vertex and axis of symmetry.
It can help you solve quadratic inequalities by finding the intervals where the function is positive or negative.
How to Find Zeros By Completing the Square in Common Core Algebra 1?
In common core algebra 1, you may encounter homework problems that ask you to find zeros by completing the square. Here are some examples and solutions:
Example 1
Solve x^2+6x-7=0 x2 +6x 7 = 0 by completing the square.
Solution:
Move the constant term to the right side: x^2+6x=7 x2 +6x = 7
Divide both sides by 1: x^2+6x=7 x2 +6x = 7 (no change)
Add the square of half of the coefficient of x x to both sides: x^2+6x+9=16 x2 +6x +9 = 16
Factor the left side as a perfect square: (x+3)^2=16 (x +3)2 = 16
Solve for x x by taking the square root of both sides and simplifying: x+3=\\pm\\sqrt16 x +3 = 16
x=-3\\pm4 x = 3 4
x=-7\\text or x=1 x = 7 or x = 1
The zeros are -7 -7 and 1 1 .
Example 2
Solve 3x^2-12x-15=0 3x2 12x 15 = 0 by completing the square.
Solution:
Move the constant term to the right side: 3x^2-12x=15 3x2 12x = 15
Divide both sides by 3: x^2-4x=5 x2 4x = 5
Add the square of half of the coefficient of x x to both sides: x^2-4x+4=9 x2 4x +4 = 9
Factor the left side as a perfect square: (x-2)^2=9 (x 2)2 = 9
Solve for x x by taking the square root of both sides and simplifying: x-2=\\pm\\sqrt9 x 2 = 9
x=2\\pm3 x = 2 3
x=-1\\text or x=5 x = 1 or x = 5
The zeros are -1 -1 and 5 5 .
Example 3
Solve \\frac 1 4x^2-\\frac 1 8x-\\frac 5 8=0 \\dfrac14
How to Find Vertex and Axis of Symmetry By Completing the Square in Common Core Algebra 1?
Completing the square can also help you find the vertex and axis of symmetry of a quadratic function. The vertex is the point where the function reaches its maximum or minimum value, and the axis of symmetry is the vertical line that passes through the vertex and divides the graph into two congruent parts.
To find the vertex and axis of symmetry by completing the square, you need to follow these steps:
Write the quadratic function in standard form: y=ax^2+bx+c y = ax2 +bx +c
Complete the square for the expression ax^2+bx ax2 +bx and factor it as a perfect square.
Write the quadratic function in vertex form: y=a(x-h)^2+k y = a(x h)2 +k , where (h,k) (h, k) is the vertex.
Find the x x -coordinate of the vertex by setting x=h x = h and solving for x x .
Find the y y -coordinate of the vertex by plugging x=h x = h into the function and solving for y y .
Find the equation of the axis of symmetry by writing x=h x = h .
Example 4
Find the vertex and axis of symmetry of the quadratic function y=2x^2-8x+5 y = 2x2 8x +5 by completing the square.
Solution:
Write the quadratic function in standard form: y=2x^2-8x+5 y = 2x2 8x +5 (no change)
Complete the square for the expression 2x^2-8x 2x2 8x and factor it as a perfect square: y=2(x^2-4x+4)-8+5 y = 2(x2 4x +4) 8 +5
y=2(x-2)^2-3 y = 2(x 2)2 3
Write the quadratic function in vertex form: y=2(x-2)^2-3 y = 2(x 2)2 3 , where (h,k)=(2,-3) (h, k) = (2, 3) is the vertex.
Find the x x -coordinate of the vertex by setting x=2 x = 2 and solving for x x : x=2 x = 2 (no change)
Find the y y -coordinate of the vertex by plugging x=2 x = 2 into the function and solving for y y : y=2(2-2)^2-3 y = 2(2 2)2 3
y=-3 y = 3
Find the equation of the axis of symmetry by writing x=2 x = 2 .
The vertex is (2,-3) (2, 3) and the axis of symmetry is x=2 x = 2 .
Example 5
Find the vertex and axis of symmetry of the quadratic function y=-\\frac 1 4x^
How to Solve Quadratic Inequalities By Completing the Square in Common Core Algebra 1?
Completing the square can also help you solve quadratic inequalities, which are expressions that involve a quadratic function and an inequality sign, such as , \\leq \\leq , or \\geq \\geq . A quadratic inequality can have one or two solutions, depending on the sign of the leading coefficient and the direction of the inequality.
To solve quadratic inequalities by completing the square, you need to follow these steps:
Write the quadratic inequality in standard form: ax^2+bx+c0 a x2 +bx +c > 0 , ax^2+bx+c\\leq0 a x2 +bx +c 0 , or ax^2+bx+c\\geq0 a x2 +bx +c 0
Complete the square for the expression ax^2+bx ax2 +bx and factor it as a perfect square.
Write the quadratic inequality in vertex form: a(x-h)^2+k0 a (x h)2 +k > 0 , a(x-h)^2+k\\leq0 a (x h)2 +k 0 , or a(x-h)^2+k\\geq0 a (x h)2 +k 0 , where (h,k) (h, k) is the vertex.
Find the zeros of the quadratic function by setting it equal to zero and solving for x x . These are the boundary points that divide the number line into intervals.
Test each interval by plugging in any value of x x from that interval into the quadratic function and checking the sign of the result. If the result satisfies the inequality, then the interval is part of the solution. If not, then the interval is not part of the solution.
Write the solution as an interval notation or a set notation, using parentheses or brackets depending on whether the boundary points are included or not.
Example 6
Solve x^2-6x+5Solution:
Write the quadratic inequality in standard form: x^2-6x+5
Complete the square for the expression x^2-6x x2 6x and factor it as a perfect square: x^2-6x+9
Write the quadratic inequality in vertex form: (x-3)^2
Find the zeros of the quadratic function by setting it equal to zero and solving for x x : (x-3)^2=4 (x 3)2 = 4
x-3=\\pm\\sqrt4 x 3 = 4
x=3\\pm2 x = 3 2
x=1\\text or x=5 x = 1 or x = 5
Test each interval by plugging in any value of x x from that interval into the quadratic function and checking the sign of the result: Interval Test value Result Solution (-\\infty,1) (-\\infty,1) -1 -1 (-1)^
How to Graph Quadratic Functions By Completing the Square in Common Core Algebra 1?
Completing the square can also help you graph quadratic functions, which are functions that have the form y=ax^2+bx+c y = ax2 +bx +c , where a a is not zero. A quadratic function has a U-shaped curve called a parabola, which can open up or down depending on the sign of a a .
To graph quadratic functions by completing the square, you need to follow these steps:
Write the quadratic function in standard form: y=ax^2+bx+c y = ax2 +bx +c
Complete the square for the expression ax^2+bx ax2 +bx and factor it as a perfect square.
Write the quadratic function in vertex form: y=a(x-h)^2+k y = a(x h)2 +k , where (h,k) (h, k) is the vertex.
Plot the vertex on the coordinate plane as a point.
Find the x x -intercepts by setting y=0 y = 0 and solving for x x . These are the points where the parabola crosses the x x -axis.
Plot the x x -intercepts on the coordinate plane as points.
Find the y y -intercept by setting x=0 x = 0 and solving for y y . This is the point where the parabola crosses the y y -axis.
Plot the y y -intercept on the coordinate plane as a point.
Draw a smooth curve that passes through all the points and opens up or down depending on the sign of a a .
Example 7
Graph the quadratic function y=x^2-4x-5 y = x2 4x 5 by completing the square.
Solution:
Write the quadratic function in standard form: y=x^2-4x-5 y = x2 4x 5 (no change)
Complete the square for the expression x^2-4x x2 4x and factor it as a perfect square: y=x^2-4x+4-9 y = x2 4x +4 9
y=(x-2)^2-9 y = (x 2)2 9
Write the quadratic function in vertex form: y=(x-2)^2-9 y = (x 2)2 9 , where (h,k)=(2,-9) (h, k) = (2, 9) is the vertex.
Plot the vertex on the coordinate plane as a point: (2,-9) (2, 9)
Find the x x -intercepts by setting y=0 y = 0 and solving for x x : 0=(x-2)^
Conclusion
Completing the square is a powerful technique that can help you solve and graph quadratic equations and functions. It can help you find the zeros, vertex, axis of symmetry, and intervals of a quadratic function. It can also help you transform a quadratic function into a perfect square form that is easy to work with. In this article, you learned how to use completing the square in common core algebra 1 homework problems and examples. By following the steps and tips in this article, you can master completing the square and ace your algebra 1 tests.
a8ba361960
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