Question

How do you solve the equation $$x^2-2x-24=0$$ by graphing?

Answer 1

Hint: Here, we will first find the $$x$$ and $$y$$ intercepts and the vertex to plot the graph of this equation. As this is a quadratic equation, its graph will be U-shaped. After plotting the graph, we will observe what are the possible values of $$x$$ when $$y=0$$ to find the required solution using the graph.

Complete step-by-step answer:
The given quadratic equation is: $$x^2-2x-24=0$$
Now, in order to plot a graph of this equation, we will, first of all, determine the $$y$$ intercept.
Hence, we will substitute $$x=0$$
Hence, $$y=f\left(0\right)={\left(0\right)}^2-2\left(0\right)-24=-24$$
Thus, the $$y$$ intercept is $$\left(0,-24\right)$$
Now, we will determine the $$x$$ intercept, by substituting $$y=f\left(x\right)=0$$
$$x^2-2x-24=0$$
Doing middle term split, we get
$$\Rightarrow x^2-6x+4x-24=0$$
$$\Rightarrow x\left(x-6\right)+4\left(x-6\right)=0$$
Factoring out common terms, we get
$$\Rightarrow \left(x+4\right)\left(x-6\right)=0$$
By using zero product property, we get
$$\begin{array}{l}\Rightarrow \left(x+4\right)=0\\ \Rightarrow x=-4\end{array}$$
Or
$$\begin{array}{l}\Rightarrow \left(x-6\right)=0\\ \Rightarrow x=6\end{array}$$
Therefore, for $$y=0$$, we get two $$x$$ intercepts, i.e. $$\left(-4,0\right)$$ and $$\left(6,0\right)$$
Now, we will determine the vertex by using the formula $$x={\displaystyle \frac{-b}{2a}}$$ to find the $$x$$-value of the vertex and then, we will substitute this value in the function to find the corresponding value of $$y$$.
In the given equation, $$x^2-2x-24=0$$ by comparing it to $$a{x}^2+bx+c=0$$, we get,
$$a=1$$, $$b=-2$$ and $$c=-24$$
Thus, substituting these values in $$x={\displaystyle \frac{-b}{2a}}$$, we get,
$$x={\displaystyle \frac{-\left(-2\right)}{2\left(1\right)}}={\displaystyle \frac{2}{2}}=1$$
Substituting this is the equation $$y=x^2-2x-24$$, we get,
$$y=1^2-2\left(1\right)-24$$
Simplifying the expression, we get
$$\Rightarrow y=1-2-24=-25$$
Therefore, the vertex is: $$\left(1,-25\right)$$
Also, let us substitute $$x=2$$ in the $$y=x^2-2x-24$$ thus, we get,
$$y=2^2-2\left(2\right)-24$$
Simplifying the expression, we get
$$\Rightarrow y=4-4-24=-24$$
Therefore, the other point is $$\left(2,-24\right)$$
Hence, we will plot the graph having:
$$y$$ intercept as $$\left(0,-24\right)$$
$$x$$ intercepts as $$\left(-4,0\right)$$ and $$\left(6,0\right)$$
Vertex: $$\left(1,-25\right)$$
Extra point: $$\left(2,-24\right)$$

Hence, from this graph, we can clearly observe that the values $$x=-4$$and $$x=6$$ when $$y=0$$

Therefore, this is the required solution of the given equation $$x^2-2x-24=0$$ by graphing.
Hence, this is the required answer.

Note:
An equation is called a quadratic equation if it can be written in the form of $$a{x}^2+bx+c=0$$ where $$a,b,c$$ are real numbers and $$a\ne 0$$ , as it is the coefficient of $$x^2$$ and it determines that this is a quadratic equation. Also, the power of a quadratic equation will be 2 as it is a ‘quadratic equation’. Also, a quadratic equation will give us two roots and its graph is U-shaped as we can observe from this question.