The quadratic formula, first discovered by the Babylonians four thousand years ago, gives you a surefire way to solve quadratic equations of the form
y = ax2 + bx + c.
Plugging in the values of a, b, and c, you will get the desired values of x.
If the expression under the square root sign (b2 – 4ac, also called the discriminant) is negative, then there are no real solutions. (You need complex numbers to deal with this case properly. These are usually taught in Algebra 2.)
If the discriminant is zero, there is only one solution. If the discriminant is positive, then the ± symbol means you get two answers.
Solve the quadratic equation.
x2– x – 12 = 0
Here a = 1, b = –1, and c = 12. Substituting, we get:
The discriminant is positive, so we have two solutions:
x = 4 and x = –3
In this example, the discriminant was 49, a perfect square, so we ended up with rational answers. Often, when using the quadratic formula, you end up with answers which still contain radicals.
3x2 + 2x + 1 = 0
Here a = 3, b = 2, and c = 1. Substituting, we get:
The discriminant is negative, so this equation has no real solutions.
Rewrite the equation 4w2 – w
= 6 in the standard form ax2 + bx + c = 0.
Compare 16x2 + 8x +1 = 0 with the standard form ax2 + bx + c = 0 and give the values of a, b and c.
Compare (x + 5)2 = 0 with the standard form ax2 + bx + c = 0 and give the values of a, b and c.
Write the equation 4p2 – 13p
= –15 + 3p in standard form ax2 + bx + c
Solve using the quadratic formula.
x2 – 6x = 16
x2 = 4x – 4
2y2 – 3y – 2 = 0
4x2 + 16x = 9
Solve the given equation using the quadratic formula.
x2 – 16 = 0
x2 – 8x + 16 = 0
y2 – 12y + 33 = 0
x2 + 6x + 5 = 6
3x2 + 10x + 5 = 0
Find the solution for the given equation by using the quadratic formula. Find the approximate answer for the irrational roots to the nearest hundredth.
a2 + 8a + 10 = 0
6y2 – y – 5 = 0
4b2 – 8b – 21 = 0
3r2 + r – 15 = 0
3x2 – 30x + 75 = 0
2x2 – 30x + 150 = 0
5r2 – 14 = –16r
Solve 16x2 + 130x + 16 = 0 by applying the quadratic formula.
Solve 6n2 + 5n = 25 by applying the quadratic formula.
Solve 5x2 – 3x + 7 = 0 by applying the quadratic formula.
You're playing ball on the moon. You throw the ball upward with a velocity of 8 meters per second. Then the height of the ball y after t seconds is given by
y = –0.8t2
At what times will it reach a height of 10m?
Graph y = –0.8t2 + 8t. Use a domain of 0 t 7.5.