The quadratic function is equivalent to . What is the value of ?

Solution available with step-by-step explanation

Super-Hard SAT Math Question 89 | Free SAT Prep | Aniko

00:00

Question 89·200 Super-Hard SAT Math Questions·Advanced Math

0 / 200

The quadratic function $a(x+4.5)^2-d$ is equivalent to $4(x+1)(x+c)$ . What is the value of $d$ ?

When two quadratic expressions are equivalent, first match easy structural features instead of expanding: (1) match the $x^2$ coefficient (leading coefficient), (2) use vertex form for the axis of symmetry and factored form for the roots (midpoint), then (3) plug in a root to make the expression 0 and solve for the remaining constant.

Hints

Click me!

Match the leading coefficient

Equivalent quadratics must have the same coefficient of $x^2$ . Compare the $x^2$ coefficients on both sides to determine $a$ .

Connect vertex form to roots

In $a(x-h)^2+k$ , the axis of symmetry is $x=h$ . In $A(x-r_1)(x-r_2)$ , the axis is the midpoint of the roots.

Use a root to eliminate $d$

Once you know $c$ , plug in a root from the factored form (like $x=-1$ ) so the expression equals 0 and you can solve for $d$ .

Desmos Guide

Enter both quadratics

Enter

$y_1=a(x+4.5)^2-d$ (sliders for $a$ and $d$ )
$y_2=4(x+1)(x+c)$ (slider for $c$ )

Use structure to set $a$ and find $c$

Since the coefficient of $x^2$ must match, set $a=4$ .

The zeros of $y_2$ are $x=-1$ and $x=-c$ , so the midpoint is $m=\frac{-1-c}{2}$ . Enter

m=\frac{-1-c}{2}

Adjust $c$ until $m=-4.5$ .

Evaluate to get $d$

At $x=-4.5$ , $y_1=4(x+4.5)^2-d$ becomes $y_1=-d$ .

Evaluate $y_2$ at $x=-4.5$ by entering:

v=y_2(-4.5)

Then $d=-v$ (which gives $49$ ).

Step-by-step Explanation

Match leading coefficients to get $a$

Because the two expressions are equivalent quadratics, they must have the same coefficient of $x^2$ .

In $a(x+4.5)^2-d$ , the coefficient of $x^2$ is $a$ .
In $4(x+1)(x+c)$ , the coefficient of $x^2$ is $4$ .

So $a=4$ . Divide both sides by 4:

(x+4.5)^2-\frac d4=(x+1)(x+c)

Use symmetry to find $c$

The left side is in vertex form, so its axis of symmetry is $x=-4.5$ .

The right side has zeros at $x=-1$ and $x=-c$ , so its axis of symmetry is the midpoint:

\frac{-1+(-c)}{2}=\frac{-1-c}{2}

Set equal to $-4.5$ and solve:

\frac{-1-c}{2}=-4.5 \Rightarrow -1-c=-9 \Rightarrow c=8.

Use a root to solve for $d$

Since $(x+1)$ is a factor, $x=-1$ makes the quadratic equal 0. Substitute into $4(x+4.5)^2-d$ :

0=4(-1+4.5)^2-d=4(3.5)^2-d=4(12.25)-d=49-d

So $d=49$ .

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation