Practice 1 Solutions

Solve. Write all answers in simplest form.

$5 x^{2} - 11 x + 6 = 0$

$\begin{matrix} a = 5, & b = - 11, & c = 6 \end{matrix} x = \frac{- (- 11) \pm \sqrt{{(- 11)}^{2} - 4 (5) (6)}}{2 (5)} x = \frac{11 \pm \sqrt{121 - 120}}{10} x = \frac{11 \pm \sqrt{1}}{10} x = \frac{11 \pm 1}{10} x = \frac{12}{10}, \frac{10}{10}$

$x = \frac{6}{5}, 1$

Note

You may choose to factor this equation if you notice that the discriminant is a perfect square.

$4 x^{2} + 8 x - 2 = x^{2} - 2 x$

$3 x^{2} + 10 x - 2 = 0 a = 3, b = 10, c = - 2 x = \frac{- (10) \pm \sqrt{{(10)}^{2} - 4 (3) (- 2)}}{2 (3)} x = \frac{- 10 \pm \sqrt{100 - (- 24)}}{6} x = \frac{- 10 \pm \sqrt{100 + 24}}{6} x = \frac{- 10 \pm \sqrt{124}}{6} x = \frac{- 10 \pm 2 \sqrt{31}}{6} x = \frac{2 (- 5 \pm \sqrt{31})}{2 (3)}$

$x = \frac{- 5 \pm \sqrt{31}}{3}$

$x^{2} + 6 x = 2$

$x^{2} + 6 x - 2 = 0 a = 1, b = 6, c = - 2 x = \frac{- (6) \pm \sqrt{{(6)}^{2} - 4 (1) (- 2)}}{2 (1)} x = \frac{- 6 \pm \sqrt{36 - (- 8)}}{2} x = \frac{- 6 \pm \sqrt{36 + 8}}{2} x = \frac{- 6 \pm \sqrt{44}}{2} x = \frac{- 6 \pm 2 \sqrt{11}}{2} x = \frac{2 (- 3 \pm \sqrt{11})}{2}$

$x = - 3 \pm \sqrt{11}$

$x^{2} + 3 x + 7 = 0$

$a = 1, b = 3, c = 7 x = \frac{- (3) \pm \sqrt{{(3)}^{2} - 4 (1) (7)}}{2 (1)} x = \frac{- 3 \pm \sqrt{9 - (28)}}{2} x = \frac{- 3 \pm \sqrt{- 19}}{2}$

$x = \frac{- 3 \pm i \sqrt{19}}{2}$

$9 x^{2} + 4 = 0$

$a = 9, b = 0, c = 4 x = \frac{- (0) \pm \sqrt{{(0)}^{2} - 4 (9) (4)}}{2 (9)} x = \frac{0 \pm \sqrt{- 144}}{18} x = \frac{\pm 12 i}{18}$

$x = \pm \frac{2 i}{3}$

Note

You may choose to solve this by using the methods learned in previous lessons: by isolating $x^{2}$ and taking the square root of both sides.

$- 2 x^{2} = - 11 x + 12$

$- 2 x^{2} + 11 x - 12 = 0 a = - 2, b = 11, c = - 12 x = \frac{- (11) \pm \sqrt{{(11)}^{2} - 4 (- 2) (- 12)}}{2 (- 2)} x = \frac{- 11 \pm \sqrt{121 - (96)}}{- 4} x = \frac{- 11 \pm \sqrt{25}}{- 4} x = \frac{- 11 \pm 5}{- 4} x = \frac{- 6}{- 4}, \frac{- 16}{- 4}$

$x = \frac{3}{2}, 4$

Note

You may have chosen to write the equation as $2 x^{2} - 11 x + 12 = 0$ , which will result in the same solution. You may also factor this equation if you notice that the discriminant is a perfect square.

$7 x^{2} - 10 x + 2 = 0$

$a = 7, b = - 10, c = 2 x = \frac{- (- 10) \pm \sqrt{{(- 10)}^{2} - 4 (7) (2)}}{2 (7)} x = \frac{10 \pm \sqrt{100 - (56)}}{14} x = \frac{10 \pm \sqrt{44}}{14} x = \frac{10 \pm 2 \sqrt{11}}{14} x = \frac{2 (5 \pm \sqrt{11})}{2 (7)}$

$x = \frac{5 \pm \sqrt{11}}{7}$

$x^{2} - 3 x - 2 = - 6$

$x^{2} - 3 x + 4 = 0 a = 1, b = - 3, c = 4 x = \frac{- (- 3) \pm \sqrt{{(- 3)}^{2} - 4 (1) (4)}}{2 (1)} x = \frac{3 \pm \sqrt{9 - (16)}}{2} x = \frac{3 \pm \sqrt{- 7}}{2}$

$x = \frac{3 \pm i \sqrt{7}}{2}$

Determine the type of roots to the quadratic equation using the discriminant. Explain.

$5 x^{2} + x + 1 = 0$

$a = 5, b = 1, c = 1 b^{2} - 4 a c {(1)}^{2} - 4 (5) (1) 1 - 20 - 19$

This equation has two complex solutions.

$5 x^{2} - 7 = 3 x^{2} + 3 x$

$2 x^{2} - 3 x - 7 = 0 a = 2, b = - 3, c = - 7 b^{2} - 4 a c {(- 3)}^{2} - 4 (2) (- 7) 9 - (- 56) 9 + 56 65$

This equation has two real, irrational solutions.

$6 x^{2} + 13 x + 2 = 0$

$a = 6, b = 13, c = 2 b^{2} - 4 a c {(13)}^{2} - 4 (6) (2) 169 - 48 121$

This equation has two real, rational solutions.

$x^{2} + 6 x + 12 = 0$

$a = 1, b = 6, c = 12 b^{2} - 4 a c {(6)}^{2} - 4 (1) (12) 36 - 48 - 12$

This equation has two complex solutions.

$3 x^{2} + 10 x = 2$

$3 x^{2} + 10 x - 2 = 0 a = 3, b = 10, c = - 2 b^{2} - 4 a c {(10)}^{2} - 4 (3) (- 2) 100 - (- 24) 100 + 24 124$

This equation has two real, irrational solutions.

$x^{2} + 4 x + 4 = 0$

$a = 1, b = 4, c = 4 b^{2} - 4 a c {(4)}^{2} - 4 (1) (4) 16 - 16 0$

This equation has one real, rational solution.

Note

This is also called a double root.

$4 x^{2} - 7 x = 3 x^{2} - 4$

$x^{2} - 7 x + 4 = 0 a = 1, b = - 7, c = 4 b^{2} - 4 a c {(- 7)}^{2} - 4 (1) (4) 49 - 16 33$

This equation has two real, irrational solutions.

$16 x^{2} + 8 x + 1 = 0$

$a = 16, b = 8, c = 1 b^{2} - 4 a c {(8)}^{2} - 4 (16) (1) 64 - 64 0$

This equation has one real, rational solution.

Cole threw a ball at an initial velocity of 35 feet per second from a starting height of 6 feet. Determine how long the ball will be in the air before it hits the ground. Round your answer to the nearest hundredth.

$0 = \frac{1}{2} (- 32) t^{2} + 35 t + 6 0 = - 16 t^{2} + 35 t + 6 a = - 16, b = 35, c = 6 t = \frac{- 35 \pm \sqrt{35^{2} - 4 (- 16) (6)}}{2 (- 16)} t = \frac{- 35 \pm \sqrt{1609}}{- 32} t = \frac{- 35 \pm 40.11}{- 32} t = - 0.16, 2.34$

The ball took 2.34 seconds to hit the ground.

Note

The negative time value is extraneous since we cannot go back in time.

Grant dropped a rock off a 20 meter cliff at an initial velocity of 0 meters per second. Find the time it takes to hit the ground in meters per second.

$\begin{array}{rcl} 0 & = & \frac{1}{2} (- 9.8) t^{2} + 0 t + 20 \\ 0 & = & - 4.9 t^{2} + 20 \\ 4.9 t^{2} & = & 20 \\ t^{2} & = & 4.08 \\ t & = & - 2.02, 2.02 \end{array}$

It takes 2.02 seconds to hit the ground.

Note

You could choose to use the quadratic formula; however, this method is more efficient.

The referee tossed a coin prior to the game. It was flipped at a velocity of 12 feet per second at a height of 5.75 feet. How much time passed until the coin hit the ground?

$\frac{1}{2} (- 32) t^{2} + 12 t + 5.75 = 0 - 16 t^{2} + 12 t + 5.75 = 0 a = - 16, b = 12, c = 5.75 t = \frac{- 12 \pm \sqrt{{(12)}^{2} - 4 (- 16) (5.75)}}{2 (- 16)} t = \frac{- 12 \pm \sqrt{512}}{- 32} t = \frac{- 12 \pm 22.63}{- 32} t = - 0.33, 1.08$

The coin hit the ground at 1.08 seconds.

Two monkeys were tossing a banana back and forth. The first monkey threw the banana at a height of 2.5 feet and the other caught it at 2 feet. The initial velocity was 25 feet per second. How long was the banana in the air for one toss?

$\frac{1}{2} (- 32) t^{2} + 25 t + 2.5 = 2 - 16 t^{2} + 25 t + 0.5 = 0 a = - 16, b = 25, c = 0.5 t = \frac{- 25 \pm \sqrt{{(25)}^{2} - 4 (- 16) (0.5)}}{2 (- 16)} t = \frac{- 25 \pm \sqrt{641}}{2 (- 16)} t = \frac{- 25 \pm 25.32}{- 32} t = - 0.01, 1.57$

The second monkey caught the banana at 1.57 seconds.