Solved on Dec 13, 2023

Rewrite $h(x)=2x^2+11x+15$ as $(x+a)^2+b$ , where $a$ and $b$ are constants.

STEP 1

Assumptions
1. We are given a quadratic function $h(x) = 2x^2 + 11x + 15$ .
2. We need to rewrite $h(x)$ in the form $h(x) = (x + a)^2 + b$ , where $a$ and $b$ are constants to be determined.
3. This process is known as completing the square.

STEP 2

To complete the square, the coefficient of $x^2$ must be 1. If it is not, we factor out the coefficient of $x^2$ from the first two terms.
$h(x) = 2(x^2 + \frac{11}{2}x) + 15$

STEP 3

Now, we find the value to complete the square for the expression inside the parentheses. This value is $\left(\frac{b}{2a}\right)^2$ , where $ax^2 + bx + c$ is the standard form of the quadratic equation.
$a = 1, b = \frac{11}{2}$
$\left(\frac{b}{2a}\right)^2 = \left(\frac{\frac{11}{2}}{2}\right)^2 = \left(\frac{11}{4}\right)^2$

STEP 4

Calculate the value to complete the square.
$\left(\frac{11}{4}\right)^2 = \left(\frac{11}{4}\right) \times \left(\frac{11}{4}\right) = \frac{121}{16}$

STEP 5

Add and subtract the calculated value inside the parentheses to complete the square. We add and subtract $\frac{121}{16}$ , but since we factored out a 2 at the beginning, we need to multiply $\frac{121}{16}$ by 2 when subtracting it outside the parentheses to maintain equality.
$h(x) = 2\left(x^2 + \frac{11}{2}x + \frac{121}{16}\right) + 15 - 2 \times \frac{121}{16}$

STEP 6

Simplify the expression by combining like terms.
$h(x) = 2\left(x + \frac{11}{4}\right)^2 + 15 - \frac{242}{16}$

STEP 7

Convert the constant terms to have a common denominator to combine them.
$15 = \frac{15}{1} = \frac{15 \times 16}{16} = \frac{240}{16}$
$h(x) = 2\left(x + \frac{11}{4}\right)^2 + \frac{240}{16} - \frac{242}{16}$

STEP 8

Combine the constant terms.
$h(x) = 2\left(x + \frac{11}{4}\right)^2 + \left(\frac{240}{16} - \frac{242}{16}\right)$
$h(x) = 2\left(x + \frac{11}{4}\right)^2 - \frac{2}{16}$

STEP 9

Simplify the constant term.
$h(x) = 2\left(x + \frac{11}{4}\right)^2 - \frac{1}{8}$

STEP 10

Finally, we rewrite the function in the desired form.
$h(x) = (x + \frac{11}{4})^2 - \frac{1}{8}$
Since we want the form $h(x) = (x + a)^2 + b$ , we can write the solution as:
$h(x) = \left(x + \frac{11}{4}\right)^2 - \frac{1}{8}$

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Rewrite $h(x)=2x^2+11x+15$ as $(x+a)^2+b$ , where $a$ and $b$ are constants.

STEP 1

Assumptions
1. We are given a quadratic function $h(x) = 2x^2 + 11x + 15$ .
2. We need to rewrite $h(x)$ in the form $h(x) = (x + a)^2 + b$ , where $a$ and $b$ are constants to be determined.
3. This process is known as completing the square.

STEP 2

To complete the square, the coefficient of $x^2$ must be 1. If it is not, we factor out the coefficient of $x^2$ from the first two terms.
$h(x) = 2(x^2 + \frac{11}{2}x) + 15$

STEP 3

STEP 4

Calculate the value to complete the square.
$\left(\frac{11}{4}\right)^2 = \left(\frac{11}{4}\right) \times \left(\frac{11}{4}\right) = \frac{121}{16}$

STEP 5

STEP 6

Simplify the expression by combining like terms.
$h(x) = 2\left(x + \frac{11}{4}\right)^2 + 15 - \frac{242}{16}$

STEP 7

Convert the constant terms to have a common denominator to combine them.
$15 = \frac{15}{1} = \frac{15 \times 16}{16} = \frac{240}{16}$
$h(x) = 2\left(x + \frac{11}{4}\right)^2 + \frac{240}{16} - \frac{242}{16}$

STEP 8

Combine the constant terms.
$h(x) = 2\left(x + \frac{11}{4}\right)^2 + \left(\frac{240}{16} - \frac{242}{16}\right)$
$h(x) = 2\left(x + \frac{11}{4}\right)^2 - \frac{2}{16}$

STEP 9

Simplify the constant term.
$h(x) = 2\left(x + \frac{11}{4}\right)^2 - \frac{1}{8}$

STEP 10

Finally, we rewrite the function in the desired form.
$h(x) = (x + \frac{11}{4})^2 - \frac{1}{8}$
Since we want the form $h(x) = (x + a)^2 + b$ , we can write the solution as:
$h(x) = \left(x + \frac{11}{4}\right)^2 - \frac{1}{8}$

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Rewrite h(x)=2x2+11x+15h(x)=2x^2+11x+15h(x)=2x2+11x+15 as (x+a)2+b(x+a)^2+b(x+a)2+b, where aaa and bbb are constants.

STEP 1

Assumptions1. We are given a quadratic function h(x)=2x2+11x+15h(x) = 2x^2 + 11x + 15h(x)=2x2+11x+15.2. We need to rewrite h(x)h(x)h(x) in the form h(x)=(x+a)2+bh(x) = (x + a)^2 + bh(x)=(x+a)2+b, where aaa and bbb are constants to be determined.3. This process is known as completing the square.

STEP 2

To complete the square, the coefficient of x2x^2x2 must be 1. If it is not, we factor out the coefficient of x2x^2x2 from the first two terms.h(x)=2(x2+112x)+15h(x) = 2(x^2 + \frac{11}{2}x) + 15h(x)=2(x2+211​x)+15

STEP 3

STEP 4

Calculate the value to complete the square.(114)2=(114)×(114)=12116\left(\frac{11}{4}\right)^2 = \left(\frac{11}{4}\right) \times \left(\frac{11}{4}\right) = \frac{121}{16}(411​)2=(411​)×(411​)=16121​

STEP 5

STEP 6

Simplify the expression by combining like terms.h(x)=2(x+114)2+15−24216h(x) = 2\left(x + \frac{11}{4}\right)^2 + 15 - \frac{242}{16}h(x)=2(x+411​)2+15−16242​

STEP 7

STEP 8

Combine the constant terms.h(x)=2(x+114)2+(24016−24216)h(x) = 2\left(x + \frac{11}{4}\right)^2 + \left(\frac{240}{16} - \frac{242}{16}\right)h(x)=2(x+411​)2+(16240​−16242​)h(x)=2(x+114)2−216h(x) = 2\left(x + \frac{11}{4}\right)^2 - \frac{2}{16}h(x)=2(x+411​)2−162​

STEP 9

Simplify the constant term.h(x)=2(x+114)2−18h(x) = 2\left(x + \frac{11}{4}\right)^2 - \frac{1}{8}h(x)=2(x+411​)2−81​

STEP 10

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Rewrite $h(x)=2x^2+11x+15$ as $(x+a)^2+b$ , where $a$ and $b$ are constants.

Assumptions
1. We are given a quadratic function $h(x) = 2x^2 + 11x + 15$ .
2. We need to rewrite $h(x)$ in the form $h(x) = (x + a)^2 + b$ , where $a$ and $b$ are constants to be determined.
3. This process is known as completing the square.

To complete the square, the coefficient of $x^2$ must be 1. If it is not, we factor out the coefficient of $x^2$ from the first two terms.
$h(x) = 2(x^2 + \frac{11}{2}x) + 15$

Calculate the value to complete the square.
$\left(\frac{11}{4}\right)^2 = \left(\frac{11}{4}\right) \times \left(\frac{11}{4}\right) = \frac{121}{16}$

Simplify the expression by combining like terms.
$h(x) = 2\left(x + \frac{11}{4}\right)^2 + 15 - \frac{242}{16}$

Combine the constant terms.
$h(x) = 2\left(x + \frac{11}{4}\right)^2 + \left(\frac{240}{16} - \frac{242}{16}\right)$
$h(x) = 2\left(x + \frac{11}{4}\right)^2 - \frac{2}{16}$

Simplify the constant term.
$h(x) = 2\left(x + \frac{11}{4}\right)^2 - \frac{1}{8}$