Solve each equation by graphing. Round to the nearest tenth.
[tex]x^2+9=-10 x[/tex]
Answer (1)
Rewrite the equation in standard quadratic form: x 2 + 10 x + 9 = 0 .
Consider the quadratic function y = x 2 + 10 x + 9 and find its roots.
The roots of the equation are approximately x = − 9 and x = − 1 .
Round the solutions to the nearest tenth: − 9.0 , − 1.0
Explanation
Rewrite the equation First, we need to rewrite the given equation x 2 + 9 = − 10 x in the standard quadratic form. To do this, we add 10 x to both sides of the equation, resulting in x 2 + 10 x + 9 = 0 .
Consider the quadratic function Now, we can think of this equation as finding the roots (x-intercepts) of the quadratic function y = x 2 + 10 x + 9 . The roots are the values of x for which y = 0 . We can find these roots by graphing the function or by using other methods such as factoring or the quadratic formula. Since the problem asks us to solve by graphing, we will find the x-intercepts from the graph.
Find the roots Using a tool, we find the approximate real-valued roots of the function x 2 + 10 x + 9 in the interval [ − 10 , 1 ] are approximately x = − 9 and x = − 1 .
State the solutions Therefore, the solutions to the equation x 2 + 10 x + 9 = 0 are x = − 9 and x = − 1 . Since the problem asks us to round to the nearest tenth, we can write these solutions as x = − 9.0 and x = − 1.0 .
Examples
Imagine you are designing a parabolic arch for a bridge. The equation x 2 + 10 x + 9 = 0 can represent the shape of the arch, where the roots of the equation indicate where the arch meets the ground. Solving this equation helps you determine the width of the base of the arch, which is crucial for structural stability. Understanding quadratic equations and their graphical solutions is essential in engineering and architecture for designing stable and efficient structures.
