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

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

Equivalent quadratic forms give the same output for every input. First match the numbers multiplying $x^2$ . Then use a zero from the factored form: it makes the output $0$ without finding the unknown in the other factor. Substitute that input into the squared form and use a Desmos regression to find the remaining constant. Keep the multiplier outside the square.

Hints

Hint 1
An identity is an equality that holds at every input, so the coefficients of $x^2$ must match. What multiplies $x^2$ in each form? Start there before working with the unknown factor.
Hint 2
A zero is an input that makes a function’s output $0$ . The factor $x+1$ vanishes at $x=-1$ , so you don’t need to find $c$ . What must the squared form equal at that input?
Hint 3
Substitute the zero into $a(x+4.5)^2-d$ , using the value you found for $a$ . The multiplier outside the square still counts; squaring the number inside the parentheses alone won’t give $d$ .

Step-by-step

Match coefficients and use a zero
Step 1
Find the multiplier
The two forms are equivalent, so their $x^2$ terms must have the same coefficient. Squaring $x+4.5$ gives an $x^2$ term multiplied by $a$ . Multiplying the $x$ terms in $4(x+1)(x+c)$ gives $4x^2$ . So $a=4$ .
Step 2
Find an input that removes the unknown factor
A zero is an input where the output is $0$ . Set the known factor $x+1$ equal to $0$ : $x+1=0$ . Subtract $1$ : $x=-1$ . At that input, $4(x+1)(x+c)$ is $0$ no matter what $c$ is. Use a known zero to avoid solving for an unknown factor.
Step 3
Use the same output in the squared form
Equivalent forms have the same output at $x=-1$ , so the squared form must equal $0$ there. Substitute $a=4$ and $x=-1$ : $0=4(-1+4.5)^2-d$ . Simplify inside the square: $0=4(3.5)^2-d$ . Don’t drop the $4$ ; it multiplies the whole square.
Step 4
Find the remaining constant
Type $0\sim4(3.5)^2-d$ in Desmos. The $\sim$ tells Desmos to fit the unknown constant; under PARAMETERS, it shows $d=49$ . So the value of $d$ is $49$ . Choice B.

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

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