Finding Zeros By Completing The Square Common Core Algebra 1 Homework Answers

Finding Zeros By Completing The Square Common Core Algebra 1 Homework Answe=
rs

Finding zeros by completing the square is a technique that can help you sol=
ve quadratic equations that are not easily factorable or have irrational ro=
ots. In this article, you will learn how to use this technique and how to a=
pply 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 taki=
ng the square root of both sides.



Finding Zeros By Completing The Square Common Core Algebra 1 Homework Answe=
rs

Download Zip https://t.co/L8JpcnnLCO=20







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, w=
hich 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 o=
f 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 solution=
s:



Example 1

Solve x^2+6x-7=3D0 x2 +6x 7 =3D 0 by completing the square.



Solution:




Move the constant term to the right side: x^2+6x=3D7 x2 +6x =3D 7
Divide both sides by 1: x^2+6x=3D7 x2 +6x =3D 7 (no change)
Add the square of half of the coefficient of x x to both sides: x^2+6x+9=3D=
16 x2 +6x +9 =3D 16
Factor the left side as a perfect square: (x+3)^2=3D16 (x +3)2 =3D 16
Solve for x x by taking the square root of both sides and simplifying: x+3=
=3D\\pm\\sqrt16 x +3 =3D  16
x=3D-3\\pm4 x =3D 3  4
x=3D-7\\text or x=3D1 x =3D 7 or x =3D 1


The zeros are -7 -7 and 1 1 .







Example 2

Solve 3x^2-12x-15=3D0 3x2 12x 15 =3D 0 by completing the square.



Solution:




Move the constant term to the right side: 3x^2-12x=3D15 3x2 12x =3D 15
Divide both sides by 3: x^2-4x=3D5 x2 4x =3D 5
Add the square of half of the coefficient of x x to both sides: x^2-4x+4=3D=
9 x2 4x +4 =3D 9
Factor the left side as a perfect square: (x-2)^2=3D9 (x 2)2 =3D 9
Solve for x x by taking the square root of both sides and simplifying: x-2=
=3D\\pm\\sqrt9 x 2 =3D  9
x=3D2\\pm3 x =3D 2  3
x=3D-1\\text or x=3D5 x =3D 1 or x =3D 5


The zeros are -1 -1 and 5 5 .



Example 3

Solve \\frac 1 4x^2-\\frac 1 8x-\\frac 5 8=3D0 \\dfrac14

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20

=20




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 symmetr=
y of a quadratic function. The vertex is the point where the function reach=
es its maximum or minimum value, and the axis of symmetry is the vertical l=
ine 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=3Dax^2+bx+c y =3D ax2 +bx =
+c
Complete the square for the expression ax^2+bx ax2 +bx and factor it as a p=
erfect square.
Write the quadratic function in vertex form: y=3Da(x-h)^2+k y =3D a(x h)2 +=
k , where (h,k) (h, k) is the vertex.
Find the x x -coordinate of the vertex by setting x=3Dh x =3D h and solving=
 for x x .
Find the y y -coordinate of the vertex by plugging x=3Dh x =3D h into the f=
unction and solving for y y .
Find the equation of the axis of symmetry by writing x=3Dh x =3D h .


Example 4

Find the vertex and axis of symmetry of the quadratic function y=3D2x^2-8x+=
5 y =3D 2x2 8x +5 by completing the square.



Solution:




Write the quadratic function in standard form: y=3D2x^2-8x+5 y =3D 2x2 8x +=
5 (no change)
Complete the square for the expression 2x^2-8x 2x2 8x and factor it as a pe=
rfect square: y=3D2(x^2-4x+4)-8+5 y =3D 2(x2 4x +4) 8 +5
y=3D2(x-2)^2-3 y =3D 2(x 2)2 3
Write the quadratic function in vertex form: y=3D2(x-2)^2-3 y =3D 2(x 2)2 3=
 , where (h,k)=3D(2,-3) (h, k) =3D (2, 3) is the vertex.
Find the x x -coordinate of the vertex by setting x=3D2 x =3D 2 and solving=
 for x x : x=3D2 x =3D 2 (no change)
Find the y y -coordinate of the vertex by plugging x=3D2 x =3D 2 into the f=
unction and solving for y y : y=3D2(2-2)^2-3 y =3D 2(2 2)2 3
y=3D-3 y =3D 3
Find the equation of the axis of symmetry by writing x=3D2 x =3D 2 .


The vertex is (2,-3) (2, 3) and the axis of symmetry is x=3D2 x =3D 2 .



Example 5

Find the vertex and axis of symmetry of the quadratic function y=3D-\\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 on=
e or two solutions, depending on the sign of the leading coefficient and th=
e direction of the inequality.



To solve quadratic inequalities by completing the square, you need to follo=
w 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 p=
erfect 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 , wh=
ere (h,k) (h, k) is the vertex.
Find the zeros of the quadratic function by setting it equal to zero and so=
lving for x x . These are the boundary points that divide the number line i=
nto 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 s=
atisfies 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 parenth=
eses or brackets depending on whether the boundary points are included or n=
ot.


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 perf=
ect 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 so=
lving for x x : (x-3)^2=3D4 (x 3)2 =3D 4
x-3=3D\\pm\\sqrt4 x 3 =3D  4
x=3D3\\pm2 x =3D 3  2
x=3D1\\text or x=3D5 x =3D 1 or x =3D 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 v=
alue Result Solution (-\\infty,1) (-\\infty,1) -1 -1 (-1)^

How to Graph Quadratic Functions By Completing the Square in Common Core Al=
gebra 1?

Completing the square can also help you graph quadratic functions, which ar=
e functions that have the form y=3Dax^2+bx+c y =3D ax2 +bx +c , where a a i=
s not zero. A quadratic function has a U-shaped curve called a parabola, wh=
ich can open up or down depending on the sign of a a .



To graph quadratic functions by completing the square, you need to follow t=
hese steps:




Write the quadratic function in standard form: y=3Dax^2+bx+c y =3D ax2 +bx =
+c
Complete the square for the expression ax^2+bx ax2 +bx and factor it as a p=
erfect square.
Write the quadratic function in vertex form: y=3Da(x-h)^2+k y =3D 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=3D0 y =3D 0 and solving for x x . The=
se 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=3D0 x =3D 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=3Dx^2-4x-5 y =3D x2 4x 5 by completing the s=
quare.



Solution:




Write the quadratic function in standard form: y=3Dx^2-4x-5 y =3D x2 4x 5 (=
no change)
Complete the square for the expression x^2-4x x2 4x and factor it as a perf=
ect square: y=3Dx^2-4x+4-9 y =3D x2 4x +4 9
y=3D(x-2)^2-9 y =3D (x 2)2 9
Write the quadratic function in vertex form: y=3D(x-2)^2-9 y =3D (x 2)2 9 ,=
 where (h,k)=3D(2,-9) (h, k) =3D (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=3D0 y =3D 0 and solving for x x : 0=
=3D(x-2)^

Conclusion

Completing the square is a powerful technique that can help you solve and g=
raph quadratic equations and functions. It can help you find the zeros, ver=
tex, axis of symmetry, and intervals of a quadratic function. It can also h=
elp you transform a quadratic function into a perfect square form that is e=
asy to work with. In this article, you learned how to use completing the sq=
uare in common core algebra 1 homework problems and examples. By following =
the steps and tips in this article, you can master completing the square an=
d ace your algebra 1 tests.

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