Which equation has solutions of 6 and -6?
A. [tex]x^2+12 x-36=0[/tex]
B. [tex]x^2-36=0[/tex]
C. [tex]x^2-12 x+36=0[/tex]
D. [tex]x^2+36=0[/tex]
Answer (1)
The problem requires finding a quadratic equation with solutions 6 and -6.
We recognize that if 6 and -6 are solutions, then (x-6) and (x+6) are factors of the quadratic.
Multiplying the factors (x-6)(x+6) gives x 2 − 36 .
Therefore, the equation is x 2 − 36 = 0 .
Explanation
Understanding the Problem We are looking for a quadratic equation that has solutions x = 6 and x = − 6 . This means that when we substitute these values into the equation, the equation should hold true.
Finding the Factors If x = 6 and x = − 6 are the solutions, then ( x − 6 ) = 0 and ( x + 6 ) = 0 . This is because if we plug in x = 6 into ( x − 6 ) we get 6 − 6 = 0 , and if we plug in x = − 6 into ( x + 6 ) we get − 6 + 6 = 0 .
Multiplying the Factors To find the quadratic equation, we multiply these two factors together: ( x − 6 ) ( x + 6 ) = 0 .
Expanding the Expression Expanding the expression, we get: x 2 − 6 x + 6 x − 36 = 0 .
Simplifying the Equation Simplifying the equation, we have: x 2 − 36 = 0 . This is because the − 6 x and + 6 x terms cancel each other out.
Final Answer Therefore, the quadratic equation with solutions 6 and -6 is x 2 − 36 = 0 .
Examples
Understanding quadratic equations and their solutions is crucial in many fields, such as physics and engineering. For example, when designing a bridge, engineers need to calculate the forces acting on the structure. These calculations often involve solving quadratic equations to determine the optimal shape and size of the bridge components. Similarly, in physics, quadratic equations are used to model projectile motion, helping to predict the trajectory of objects thrown into the air. Mastering the relationship between solutions and equations empowers you to tackle real-world problems with confidence.
