Quadratic equations: flashcards, quiz and summary

Flashcards

1. 1Why must the equation equal 0 before you factor it?

   The zero product property only works when a product equals 0. If (x − 2)(x − 3) = 0, then one factor must be 0. If the equation is not set to 0 first, you cannot conclude anything about the values of the factors.

2. 2Why do square-root solutions come in pairs?

   Both a positive and a negative number square to the same positive result, so x² = 49 gives x = 7 or x = −7. Forgetting the negative root loses half of the solutions.

3. 3How does completing the square turn x² + bx into a perfect square?

   You add (b/2)², the square of half the coefficient of x. For x² + 6x, you add 9 to get x² + 6x + 9 = (x + 3)², and then take the square root of both sides.

4. 4What does the discriminant tell you about the solutions of a quadratic?

   If b² − 4ac is positive, there are two real solutions. If it is zero, there is one double root. If it is negative, there are no real solutions, and the parabola never crosses the x-axis.

5. 5Why does the quadratic formula work for every quadratic equation?

   It is derived by completing the square on the general equation ax² + bx + c = 0, so it applies to any values of a, b, and c with a not equal to 0. Factoring can be hard when the numbers do not split nicely, but the formula still gives the answers.

6. 6How do you find the vertex of y = ax² + bx + c?

   Find the axis of symmetry with x = −b/(2a). Then substitute that x value into the equation to get y. The vertex is the point (h, k) found this way.

7. 7What does the sign of a tell you about a parabola?

   When a > 0, the parabola opens up. When a < 0, it opens down.

8. 8How are the roots of a quadratic related to its graph?

   The roots are the x values where the parabola crosses the x-axis. They are also called zeros or x-intercepts. A quadratic has at most two real roots, so its graph crosses the x-axis at most twice.

9. 9When is completing the square a better choice than the quadratic formula?

   Completing the square is better when you need the vertex or vertex form, because it produces y = a(x − h)² + k directly. For just solving, the quadratic formula is usually the most reliable route when factoring is not obvious.

10. 10What does a perfect-square discriminant tell you?

    It tells you the roots are rational, so the quadratic can usually be factored. Checking this is a quick step before you try factoring.

Practice quiz

Pick an answer to check it.

1. 1What are the solutions to x² − 5x + 6 = 0?

   1. x = −2 or x = −3
   2. x = 2 or x = 3
   3. x = −1 or x = −6
   4. x = 1 or x = 6
   Show the answer

   B. x = 2 or x = 3 The factors (x − 2) and (x − 3) multiply to x² − 5x + 6. Setting each factor to 0 gives x = 2 or x = 3. The choices with negative numbers come from flipping the signs of the factors.

2. 2What are all the solutions to x² = 49?

   1. x = 7 only
   2. x = 24.5
   3. x = 7 or x = −7
   4. x = −7 only
   Show the answer

   C. x = 7 or x = −7 Both 7 and −7 square to 49, so a square-root solution comes as a ± pair. Writing only one root loses a solution.

3. 3A quadratic has discriminant b² − 4ac = −56. How many real solutions does it have?

   1. No real solutions
   2. One real solution, a double root
   3. Two real solutions
   4. Two rational solutions
   Show the answer

   A. No real solutions A negative discriminant means the parabola never crosses the x-axis, so there are no real solutions. Two rational solutions would require a positive perfect-square discriminant.

4. 4For y = x² + 6x + 5, what is the vertex?

   1. (−6, 5)
   2. (−3, −4)
   3. (3, −4)
   4. (−3, 4)
   Show the answer

   B. (−3, −4) The axis of symmetry is x = −b/(2a) = −6/2 = −3. Substituting gives y = 9 − 18 + 5 = −4. Flipping the sign of x gives (3, −4), and skipping the 2 in 2a gives x = −6.

5. 5What are the solutions to x² + x − 6 = 0 using the quadratic formula?

   1. x = 2 or x = 3
   2. x = −2 or x = 3
   3. x = 4 or x = −6
   4. x = 2 or x = −3
   Show the answer

   D. x = 2 or x = −3 Here b² − 4ac = 1 + 24 = 25, so x = (−1 ± 5) / 2, which gives x = 2 or x = −3. Forgetting to divide by 2a gives the 4 and −6 pair, and dropping the sign on b gives the reversed pair.

6. 6To complete the square for x² + 6x = 7, what number do you add to both sides?

   1. 3
   2. 6
   3. 18
   4. 9
   Show the answer

   D. 9 Take half of 6 to get 3, then square it to get 9. Adding 9 gives (x + 3)² = 16, so x = 1 or x = −7. Adding 3 or 6 does not make a perfect square.

Test traps

What teachers like to test on this, and the mistakes students make.

• Trying to factor before setting the equation equal to 0. The zero product property only works when the product equals 0.
• Dropping the negative root when taking a square root, so x² = 49 gets only x = 7 instead of x = 7 or x = −7.
• Sign errors with −b when b is negative. If b = −4, then −b = 4, not −4.
• Dividing only part of the numerator by 2a in the quadratic formula, which is a common cause of wrong answers.
• Writing real answers when the discriminant b² − 4ac is negative. A negative discriminant means no real solutions.
• Confusing the x coordinate of the vertex, −b/(2a), with the y value. You must substitute x back into the equation to find y.

Summary

• A quadratic is an equation in the form ax² + bx + c = 0 with a not equal to 0, and its graph is a parabola.
• The solutions, or roots, are the x values where the parabola crosses the x-axis, and a quadratic has at most 2 real solutions.
• Factoring is the fastest method when the expression splits easily, but the equation must equal 0 before you factor.
• Square roots work when there is no bx term, and you must include both the positive and the negative root.
• Completing the square works for any quadratic and also turns standard form into vertex form.
• The quadratic formula x = (−b ± √(b² − 4ac)) / (2a) works for every quadratic.
• The discriminant b² − 4ac decides whether there are two, one, or no real solutions.
• The vertex lies on the axis of symmetry x = −b/(2a), and you find its y value by substituting that x back into the equation.

Key terms

Standard form

The form ax² + bx + c = 0, where a is not 0.

Parabola

The U-shaped graph of y = ax² + bx + c. It opens up when a > 0 and down when a < 0.

Root (zero, x-intercept)

An x value where the parabola crosses the x-axis, so that the quadratic equals 0 there.

Zero product property

If a product of factors equals 0, then at least one factor must equal 0. This is why each factor is set to 0 after factoring.

Discriminant

The expression b² − 4ac. Its sign tells you how many real solutions the quadratic has.

Double root

A single real solution that occurs when the discriminant is 0. The vertex touches the x-axis at this point.

Axis of symmetry

The vertical line x = −b/(2a) that passes through the vertex of the parabola.

Vertex form

The form y = a(x − h)² + k, where (h, k) is the vertex.

Completing the square

Adding (b/2)² to both sides of an equation so that x² + bx becomes a perfect square.

How to study it

The notes this pack came from

Questions