Q11. - Algebra 1

Question

Record your answers on a sheet of paper. Show your work.

Use the equation and its graph to answer each question.

a. Factor $x^2-7x+10$

b. What are the solutions of $x^2-7x+10=0$?

c. What do you notice about the graph of the quadratic equation and where it crosses the x-axis? How do these values compare to the solutions of $x^2-7x+10=0$? Explain.

Step-by-Step Solution

Answer

a. The factorisation of $x^2-7x+10$ is $\left(x-5\right)\left(x-2\right)$.

b. The solutions of $x^2-7x+10$ are $x=5$ and $x=2$.

c. It can be noticed that graph of the given quadratic function crosses the x-axis at $x=2$ and $x=5$ and the solutions of the given quadratic equation are $x=2$ and $x=5$ which is suggesting that the points at which graph crosses the x-axis are also the solutions of the given quadratic equation.

Part a. Step 1. Write the expression x 2 − 7 x + 10 by splitting the middle term.

The expression $x^2-7x+10$ by splitting the middle term can be written as:

$x^2-7x+10=x^2-5x-2x+10$

Part a. Step 2. Write the factorisation of the expression x 2 − 7 x + 10 .

The factorisation of the expression $x^2-7x+10$ is:

$$\begin{gathered} x^2-7x+10=x^2-5x-2x+10 \\ =x\left(x-5\right)-2\left(x-5\right) \\ =\left(x-5\right)\left(x-2\right) \end{gathered}$$

Therefore, the factorisation of $x^2-7x+10$ is $\left(x-5\right)\left(x-2\right)$.

Part b. Step 1. Write the zero product property.

The zero product property states that if the product of two factors is 0, then at least one of the factors must be 0.

That implies for any real numbers a and b, if $ab=0$, then $a=0$, $b=0$, or both a and b equal zero.

Part b. Step 2. Find the solutions of x 2 − 7 x + 10 = 0 .

The solutions of the equation $x^2-7x+10=0$ is:

$$\begin{gathered} x^2-7x+10=0 \\ \left(x-5\right)\left(x-2\right)=0 \\ x-5=0\text{ or }x-2=0\left(\text{zero product property}\right) \\ x=5\text{ or }x=2 \end{gathered}$$

Therefore, the solutions of $x^2-7x+10$ are $x=5$ and $x=2$.

Part c. Step 1. Observe the graph of the quadratic function.

The given graph is:

From the given graph, it can be noticed that the graph of the quadratic function crosses the x-axis at the points $\left(2,0\right)$ and $\left(5,0\right)$.

Part c. Step 2. Determine what is noticed about the graph of the quadratic equation and where it crosses the x -axis.

It can be noticed that graph of the given quadratic function crosses the x-axis at $x=2$ and $x=5$ and the solutions of the given quadratic equation are $x=2$ and $x=5$ which is suggesting that the points at which graph crosses the x-axis are also the solutions of the given quadratic equation.