### All PSAT Math Resources

## Example Questions

### Example Question #301 : Equations / Inequalities

The formula to solve a quadratic expression is:

All of the following equations have real solutions EXCEPT:

**Possible Answers:**

**Correct answer:**

We can use the quadratic formula to find the solutions to quadratic equations in the form ax^{2 }+ bx + c = 0. The quadratic formula is given below.

In order for the formula to give us real solutions, the value under the square root, b^{2 }– 4ac, must be greater than or equal to zero. Otherwise, the formula will require us to find the square root of a negative number, which gives an imaginary (non-real) result.

In other words, we need to look at each equation and determine the value of b^{2 }–^{ }4ac. If the value of b^{2 }– 4ac is negative, then this equation will not have real solutions.

Let's look at the equation 2x^{2} – 4x + 5 = 0 and determine the value of b^{2 }– 4ac.

b^{2 }– 4ac = (–4)^{2} – 4(2)(5) = 16 – 40 = –24 < 0

Because the value of b^{2 }– 4ac is less than zero, this equation will not have real solutions. Our answer will be 2x^{2} – 4x + 5 = 0.

If we inspect all of the other answer choices, we will find positive values for b^{2 }– 4ac, and thus these other equations will have real solutions.

### Example Question #22 : How To Find The Solution To A Quadratic Equation

Let , and let . What is the sum of the possible values of such that .

**Possible Answers:**

**Correct answer:**

We are told that f(x) = x^{2} - 4x + 2, and g(x) = 6 - x. Let's find expressions for f(k) and g(k).

f(k) = k^{2} – 4k + 2

g(k) = 6 – k

Now, we can set these expressions equal.

f(k) = g(k)

k^{2} – 4k +2 = 6 – k

Add k to both sides.

k^{2} – 3k + 2 = 6

Then subtract 6 from both sides.

k^{2} – 3k – 4 = 0

Factor the quadratic equation. We must think of two numbers that multiply to give us -4 and that add to give us -3. These two numbers are –4 and 1.

(k – 4)(k + 1) = 0

Now we set each factor equal to 0 and solve for k.

k – 4 = 0

k = 4

k + 1 = 0

k = –1

The two possible values of k are -1 and 4. The question asks us to find their sum, which is 3.

The answer is 3.

### Example Question #21 : Quadratic Equations

Note: Figure NOT drawn to scale.

Refer to the above diagram, which shows Rectangle with .

is the midpoint of ; ;

Evaluate (to the nearest tenth, if applicable).

**Possible Answers:**

Insufficient information is given to answer the question.

**Correct answer:**

The corresponding sides of similar triangles are in proportion, so we can set up and solve the proportion statement for :

, so

For the sake of simplicuty, we will let

Since is the midpoint of , .

Also, .

The proportion statement becomes

Solve for using cross-products:

By the quadratic equation, setting :

There are two possibilities:

or

is divided into segments of length 2.9 and 17.1. The lesser is the length of , so the correct choice is 2.9.

### Example Question #21 : Quadratic Equations

Solve for :

**Possible Answers:**

**Correct answer:**

Begin by distributing the three on the right side of the equation:

Next combine your like terms by subtracting from both sides to give you

Next, subtract 9 from both sides to give you . To solve for , now take the square root of both sides. This gives you the answer,

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