Lesson 10-4

Solving Quadratic Equations by Using the Quadratic Formula

Key ConceptThe solutions of a quadratic equation in the form of ax2 + bx + c = 0 where a 0, are given by the Quadratic Formula.

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x =−b ± b2 − 4ac

2a

Methods to solve Quadratic Equations

Method Can Be Used Comments

Graphing Always Not always exact; use only when an approximate solution is sufficient.

Factoring Sometimes Use if constant term is 0 or factors are easily determined.

Completing

The Square

Always Useful for equations of the form

x2 + bx + c = 0, where b is an even number.

Quadratic

Formula

Always Other methods may be easier to use in some cases but this method always gives accurate solutions.

Discriminant Negative Zero Positive

Example 2x2 + x + 3 = 0

There are no roots since no real number can be the square root of a negative number.

x2 + 6x + 9 = 0

There is a double root, -3

x2 - 5x + 2 = 0

There are two roots,

Number of

Real Roots

0 1 2

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x =−1± 12 − 4(2)(3)

2(2)

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x =−1± −23

4

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x =−6 ± 62 − 4(1)(9)

2(1)

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x =−6 ± 0

2

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=−62or − 3

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x =−(−5) ± (−5)2 − 4(1)(2)

2(1)

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x =5 ± 17

2

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x =5 + 17

2and

5 − 17

2

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Graphs

Use two methods to solve x2 - 2x -35 = 0.

{-5, 7}

Solve 15x2 -8x = 4 by using the Quadratic Formula. Round to the nearest tenth if necessary.

{-0.3, 0.8}

Two possible future destinations of astronauts are the planet Mars and a moon of the planet Jupiter, Europa. The gravitational acceleration on Mars is about 3.7 meters per second squared and on Europa, it is only 1.3 meters per second squared. Using this equation, (H = -1/2gt2 + vt + h, where g is gravitational pull, v is initial velocity, and h is initial height), to find how much longer baseballs thrown on Mars and on Europa will stay above the ground than a similarly thrown baseball on Earth. The initial velocity (v) is 10 meters per second and the ball is let go 2 meters above the ground (h). On Earth, the ball will stay in the air about 2.2 seconds.

{Mars 3.4 seconds longer

Europa 13.4 seconds longer}

State the value of the discriminant for each equation. Then determine the number of real roots of the equation.

4x2 - 2x + 14 = 0 The discriminant is -220, so there is no real root.

x2 + 24x = -144 The discriminant is 0 so the equation has one real root.

3x2 + 10x = 12 The discriminant is 244 so the equation has two real roots.

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