Which of the following values are the solutions of $x^2+20=12 x$? Choose two correct answers.
The given equation is \(x^2 + 20 = 12x\). To find its solutions, we first express it in the standard quadratic form \(ax^2 + bx + c = 0\). By subtracting \(12x\) from both sides of the equation, we get:
\[x^2 - 12x + 20 = 0\]
This is a standard quadratic equation where the coefficients are \(a = 1\), \(b = -12\), and \(c = 20\). Our goal is to determine the values of \(x\) that satisfy this equation.
To solve the quadratic equation \(x^2 - 12x + 20 = 0\), we look for two numbers that multiply to the constant term \(20\) and add up to the coefficient of the linear term \(-12\).
Checking the factors of \(20\):
The pair that works is \(-2\) and \(-10\). Thus, we can factor the quadratic expression as:
\[(x - 2)(x - 10) = 0\]
Setting each factor equal to zero, we find the roots of the equation:
Comparing these results to the given options:
The correct solutions are \(x = 2\) and \(x = 10\), which correspond to options (a) and (b).
Quadratic equations are fundamental in modeling real-world trajectories. For instance, if you throw a ball, its height \(h\) at time \(t\) can be modeled by a quadratic equation like \(h(t) = -5t^2 + v_0t + h_0\). Solving for \(h(t) = 0\) helps you determine exactly when the ball hits the ground. Understanding how to factor or use the quadratic formula allows you to predict these critical moments accurately in physics, engineering, and sports science.