Questions: Which of these choices is a solution to the equation? Solve by graphing. (x-6)(x+5)=-10 a) -5 b) -4 c) 3 d) -2

Solution Steps

To solve the equation \((x-6)(x+5)=-10\) by graphing, we can first rewrite the equation as \((x-6)(x+5) + 10 = 0\). This allows us to consider the function \(f(x) = (x-6)(x+5) + 10\) and find the x-values where \(f(x) = 0\). We can plot this function and identify the x-values where the graph intersects the x-axis. These x-values are the solutions to the equation.

The given equation is:

\[ (x-6)(x+5) = -10 \]

To solve this by graphing, we first rewrite it in the standard form of a quadratic equation. Expand the left side:

\[ (x-6)(x+5) = x^2 + 5x - 6x - 30 = x^2 - x - 30 \]

Now, set the equation equal to zero:

\[ x^2 - x - 30 + 10 = 0 \quad \Rightarrow \quad x^2 - x - 20 = 0 \]

The equation \(x^2 - x - 20 = 0\) can be represented by the quadratic function:

\[ f(x) = x^2 - x - 20 \]

Graph this function to find the x-values where the function equals zero. These x-values are the solutions to the equation.

By graphing \(f(x) = x^2 - x - 20\), we look for the x-intercepts, which are the points where the graph crosses the x-axis. These points are the solutions to the equation.

To verify, substitute each choice into the original equation to see if it satisfies the equation:

• For \(x = -5\):

  \[ (-5-6)(-5+5) = (-11)(0) = 0 \neq -10 \]

• For \(x = -4\):

  \[ (-4-6)(-4+5) = (-10)(1) = -10 \]

• For \(x = 3\):

  \[ (3-6)(3+5) = (-3)(8) = -24 \neq -10 \]

• For \(x = -2\):

  \[ (-2-6)(-2+5) = (-8)(3) = -24 \neq -10 \]

Final Answer