Use factoring to solve the quadratic equation. Check by substitution or by using a graphing utility and identifying $x$-intercepts.
$5 x^2=8 x+4$
Rewrite the equation in factored form.
$\square$ $=0$
(Factor completely.)
Answer (2)
To solve the quadratic equation 5 x 2 = 8 x + 4 by factoring:
Rewrite the equation in standard form: 5 x 2 − 8 x − 4 = 0 .
Factor the quadratic expression: ( 5 x + 2 ) ( x − 2 ) = 0 .
Set each factor equal to zero and solve for x : 5 x + 2 = 0 or x − 2 = 0 .
The solutions are x = − 5 2 and x = 2 .
x = − 5 2 , 2
Explanation
Rewrite the equation First, we need to rewrite the given quadratic equation 5 x 2 = 8 x + 4 in the standard form a x 2 + b x + c = 0 . Subtracting 8 x and 4 from both sides, we get:
Standard form 5 x 2 − 8 x − 4 = 0
Find two numbers Now, we need to factor the quadratic expression 5 x 2 − 8 x − 4 . We are looking for two numbers that multiply to 5 × ( − 4 ) = − 20 and add up to − 8 . These numbers are − 10 and 2 .
Rewrite the middle term Rewrite the middle term using these numbers: 5 x 2 − 10 x + 2 x − 4 = 0 .
Factor by grouping Factor by grouping: 5 x ( x − 2 ) + 2 ( x − 2 ) = 0 .
Factor out the common factor Factor out the common factor ( x − 2 ) : ( 5 x + 2 ) ( x − 2 ) = 0 .
Factored form Now we have the equation in factored form: ( 5 x + 2 ) ( x − 2 ) = 0 .
Set each factor to zero To find the solutions, we set each factor equal to zero and solve for x :
5 x + 2 = 0 or x − 2 = 0
Solve for x Solving for x :
5 x = − 2 ⟹ x = − 5 2
x = 2
Solutions So the solutions are x = − 5 2 and x = 2 .
Check the solutions We can check the solutions by substituting them back into the original equation. Let's check x = − 5 2 :
5 ( − 5 2 ) 2 = 5 ( 25 4 ) = 5 4
8 ( − 5 2 ) + 4 = − 5 16 + 5 20 = 5 4
Since both sides are equal, x = − 5 2 is a solution.
Now let's check x = 2 :
5 ( 2 ) 2 = 5 ( 4 ) = 20
8 ( 2 ) + 4 = 16 + 4 = 20
Since both sides are equal, x = 2 is a solution.
Examples
Factoring quadratic equations is a fundamental skill in algebra and has many real-world applications. For example, engineers use quadratic equations to model the trajectory of projectiles, such as the path of a ball thrown in the air or the flight of a rocket. By factoring the quadratic equation that represents the projectile's height as a function of time, they can determine when the projectile will hit the ground or reach its maximum height. This information is crucial for designing safe and efficient systems.
The quadratic equation 5 x 2 − 8 x − 4 = 0 factors to ( 5 x + 2 ) ( x − 2 ) = 0 . The solutions are x = − 5 2 and x = 2 , which can be verified by substituting back into the original equation. Both solutions are valid as they satisfy the equation.
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