It may appear that there is only one factor in the next example. Remember, however, that
means
.
Solve:
.
Solution
Rewrite the left side as a product.
Use the Zero Product Property and
set each factor to 0.
Solve the equations.
When a solution repeats, we call it
a double root.
Check your answer.
This image shows the steps for solving the equation (y – 8) squared = 0. The first step is to write the left hand side as a product, (y – 8)(y – 8) = 0. The next step is using the zero product property and set each factor equal to 0, y – 8 = 0 and y – 8 -= 0. Solve both equations, y = 8 and y = 8. When the solution repeats, it is a double root. Finally, check the solution by substituting back into the original equation.
Each of the equations we have solved in this section so far had one side in factored form. In order to use the Zero Product Property, the quadratic equation must be factored, with zero on one side. So we be sure to start with the quadratic equation in standard form,
. Then we factor the expression on the left.
Before we factor, we must make sure the quadratic equation is in standard form.
Solve:
.
Solution
Write the quadratic equation in standard form.
Factor the quadratic expression.
Use the Zero Product Property
to set each factor to 0.
Solve each equation.
Check your answers.
This image shows the steps for solving the equation 2 y squared = 13 y + 45. The first step is writing the equation in standard quadratic form, 2 y squared – 13 y – 45 = 0. The second step is to factor the quadratic expression, (2 y + 5)(y – 9) = 0. The third step is to use the zero product property to set each factor equal to 0, 2 y + 5 = 0 or y – 9 = 0. Solve each equation, y = −5/2 or y = 9. Finally, check the answers by substituting them back into the original equation.
Use the Zero Product Property
to set each factor to 0.
Solve each equation.
Check your answers.
This image shows the steps for solving the equation 5 x squared – 13 x = 7 x. The first step is writing the equation in standard quadratic form, 5 x squared – 20 x = 0. The second step is to factor the quadratic expression, 5 x(x – 4)= 0. The third step is to use the zero product property to set each factor equal to 0, 5 x = 0 or x − 4 = 0. Solve each equation, x = 0 or x = 4. Finally, check the answers by substituting them back into the original equation.
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
from theory: distance [miles] = speed [mph] × time [hours]
info #1
speed_Dennis × 1.5 = speed_Wayne × 2
=> speed_Wayne = 0.75 × speed_Dennis (i)
info #2
speed_Dennis = speed_Wayne + 7 [mph] (ii)
use (i) in (ii) => [...]
speed_Dennis = 28 mph
speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5.
Substituting the first equation into the second:
W * 2 = (W + 7) * 1.5
W * 2 = W * 1.5 + 7 * 1.5
0.5 * W = 7 * 1.5
W = 7 * 3 or 21
W is 21
D = W + 7
D = 21 + 7
D = 28
Salma
Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67.
Check:
Sales = 3542
Commission 12%=425.04
Pay = 500 + 425.04 = 925.04.
925.04 > 925.00
Munster
difference between rational and irrational numbers
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
