What are the solutions of $8 x^2=6+22 x$ ? Choose two correct answers.
A. $x=3$
B. $x=-3$
C. $x=-\frac{1}{4}$
D. $x=6
Answer (1)
Rewrite the equation in standard form: 8 x 2 − 22 x − 6 = 0 .
Simplify the equation by dividing by 2: 4 x 2 − 11 x − 3 = 0 .
Factor the quadratic equation: ( 4 x + 1 ) ( x − 3 ) = 0 .
Solve for x : x = − 4 1 and x = 3 . Thus, the solutions are x = 3 , x = − 4 1 .
Explanation
Rewrite the equation First, we need to rewrite the given equation 8 x 2 = 6 + 22 x in the standard quadratic form, which is a x 2 + b x + c = 0 . Subtracting 6 + 22 x from both sides, we get 8 x 2 − 22 x − 6 = 0 .
Simplify and prepare to factor Now, we can try to factor the quadratic equation. Notice that all coefficients are even, so we can divide the entire equation by 2 to simplify it: 4 x 2 − 11 x − 3 = 0 . We are looking for two numbers that multiply to 4 × − 3 = − 12 and add up to − 11 . These numbers are − 12 and 1 . So we can rewrite the middle term as − 12 x + x : 4 x 2 − 12 x + x − 3 = 0 .
Factor the quadratic Now, we factor by grouping: 4 x ( x − 3 ) + 1 ( x − 3 ) = 0 . This gives us ( 4 x + 1 ) ( x − 3 ) = 0 .
Solve for x Setting each factor equal to zero, we have 4 x + 1 = 0 or x − 3 = 0 . Solving for x , we get x = − 4 1 or x = 3 .
Final Answer The solutions are x = − 4 1 and x = 3 . Checking the given options, we see that these are indeed two of the correct answers.
Examples
Quadratic equations are incredibly useful in various real-world scenarios. For instance, they can model the trajectory of a ball thrown in the air, helping to determine its maximum height and range. They're also used in engineering to design bridges and buildings, ensuring structural stability. Moreover, quadratic equations play a crucial role in economics, where they can model cost and revenue curves to optimize profits. Understanding how to solve quadratic equations, especially by factoring, provides a foundation for tackling many practical problems.
