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PSAT Math : Algebra

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Example Questions

Example Question #1 : Quadratic Equations

The difference between a number and its square is 72. What is the number?

Possible Answers:

Correct answer:

Explanation:

x² – x = 72. Solve for x using the quadratic formula and x = 9 and –8. Only 9 satisfies the restrictions.

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Example Question #1 : Quadratic Equations

Given $x^{2}-6x+9=0$ and $x\cdot k=1$ , find the value of $k$ .

Possible Answers:

$\frac{1}{2}$

$\frac{2}{3}$

$\frac{3}{2}$

$1$

$\frac{1}{3}$

Correct answer:

$\frac{1}{3}$

Explanation:

We can factor the quadratic equation into $(x-3)^{2}=0$ .

Then we can see that $x=3$ .

Therefore, $x\cdot k=1$ becomes $3\cdot k=1$ and $k=\frac{1}{3}$ .

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Example Question #1 : Quadratic Equations

Which of the following is a root of the function $f(x)=2x^2-7x-4$ ?

Possible Answers:

$x = \frac{1}{4}$

$x = -\frac {1}{2}$

$x = \frac{1}{2}$

$x = -2$

$x = -4$

Correct answer:

$x = -\frac {1}{2}$

Explanation:

The roots of a function are the x intercepts of the function. Whenever a function passes through a point on the x-axis, the value of the function is zero. In other words, to find the roots of a function, we must set the function equal to zero and solve for the possible values of x.

$f(x)=2x^2-7x-4$ $= 0$

This is a quadratic trinomial. Let's see if we can factor it. (We could use the quadratic formula, but it's easier to factor when we can.)

Because the coefficient in front of the $x^2$ is not equal to 1, we need to multiply this coefficient by the constant, which is –4. When we mutiply 2 and –4, we get –8. We must now think of two numbers that will multiply to give us –8, but will add to give us –7 (the coefficient in front of the x term). Those two numbers which multiply to give –8 and add to give –7 are –8 and 1. We will now rewrite –7x as –8x + x.

$2x^2-7x-4=2x^2-8x+x-4=0$

We will then group the first two terms and the last two terms.

$(2x^2-8x)+(x-4)=0$

We will next factor out a 2x from the first two terms.

$(2x^2-8x)+(x-4)=2x(x-4)+1(x-4)=(2x+1)(x-4)=0$

Thus, when factored, the original equation becomes (2x + 1)(x – 4) = 0.

We now set each factor equal to zero and solve for x.

$2x + 1 = 0$

Subtract 1 from both sides.

2x = –1

Divide both sides by 2.

$x=-\frac{1}{2}$

Now, we set x – 4 equal to 0.

x – 4 = 0

Add 4 to both sides.

x = 4

The roots of f(x) occur where x = $-\frac{1}{2},4$ .

The answer is therefore $x = -\frac {1}{2}$ .

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Example Question #1 : Quadratic Equations

36x² -12x - 15 = 0

Solve for x

Possible Answers:

1/2 and 5/6

-1/2 and -5/6

-1/2 and 5/6

1/2 and -1/3

1/2 and 1/3

Correct answer:

-1/2 and 5/6

Explanation:

36x² - 12x - 15 = 0

Factor the equation:

(6x + 3)(6x - 5) = 0

Set each side equal to zero

6x + 3 = 0

x = -3/6 = -1/2

6x – 5 = 0

x = 5/6

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Example Question #1 : How To Factor The Quadratic Equation

$\textup{If }x-y=5\textup{ and }x^{2}-y^{2}=35,\textup{ then }xy=$

Possible Answers:

Correct answer:

Explanation:

$x^{2}-y^{2}=(x-y)(x+y)\textup{ Substitute. Given: }x-y = 5,\textup{ so:}$

$5(x+y)=35\:,\:x+y = 7 ,\: \textup{Combine equations: } x+y+x-y=7+5$

$2x = 12\:,\:x=6,\:y=1,\: \:(6)(1)=6$

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Example Question #1 : How To Factor The Quadratic Equation

Factor the following:

x^2+11x+30

Possible Answers:

(x + 3)(x+10)

(x + 5)(x+6)

(x - 5)(x-6)

(x + 15)(x+2)

(x +30)(x-1)

Correct answer:

(x + 5)(x+6)

Explanation:

Since you have an addition for the last element, you know that you will have either two subtractions or two additions in your factored groups. Since the second term is positive, your factored groups will contain two additions:

(x+?)(x+?)

Now, all you have to do are go through the factors of 30 and find the pair that adds up to 11:

and : No!

and : No! (But closer!)

and : Yes!

Therefore, your answer will be:

(x + 5)(x+6)

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Example Question #261 : Algebra

If f(x) = x² – 5 for all values x and f(a) = 4, what is one possible value of a?

Possible Answers:

Correct answer:

Explanation:

Using the defined function, f(a) will produce the same result when substituted for x:

f(a) = a² – 5

Setting this equal to 4, you can solve for a:

a² – 5 = 4

a² = 9

a = –3 or 3

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Example Question #2 : Algebraic Functions

If the function g is defined by g(x) = 4x + 5, then 2g(x) – 3 =

Possible Answers:

4x + 2

6x + 7

8x + 7

8x + 2

6x + 2

Correct answer:

8x + 7

Explanation:

The function g(x) is equal to 4x + 5, and the notation 2g(x) asks us to multiply the entire function by 2. 2(4x + 5) = 8x + 10. We then subtract 3, the second part of the new equation, to get 8x + 7.

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Example Question #2 : How To Find F(X)

If f(x) = x² + 5x and g(x) = 2, what is f(g(4))?

Possible Answers:

Correct answer:

Explanation:

First you must find what g(4) is. The definition of g(x) tells you that the function is always equal to 2, regardless of what “x” is. Plugging 2 into f(x), we get 2² + 5(2) = 14.

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Example Question #262 : Algebra

f(a) = ¹/₃(a³ + 5a – 15)

Find a = 3.

Possible Answers:

Correct answer:

Explanation:

Substitute 3 for all a.

(1/3) * (3³ + 5(3) – 15)

(1/3) * (27 + 15 – 15)

(1/3) * (27) = 9

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PSAT Math : Algebra

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : Quadratic Equations

Example Question #1 : Quadratic Equations

Example Question #1 : Quadratic Equations

Example Question #1 : Quadratic Equations

Example Question #1 : How To Factor The Quadratic Equation

Example Question #1 : How To Factor The Quadratic Equation

Example Question #261 : Algebra

Example Question #2 : Algebraic Functions

Example Question #2 : How To Find F(X)

Example Question #262 : Algebra

All PSAT Math Resources

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