91Ó°ÊÓ

In the following exercises, determine the number of solutions for each quadratic equation.

$$\begin{gathered} \left(a\right)25p^2+10p+1=0 \\ \left(b\right)7q^2-3q-6=0 \\ \left(c\right)7y^2+2y+8=0 \end{gathered}$$

$25p^2+10p+1=0$

The equation is in standard form, identify a, b, and c.

$a=25,b=10,c=1$

Write the discriminant.

$b^2-4ac$

Substitute in the values of a, b, and c and simplify.

${10}^2-4·25·1=100-100=0$

Since the discriminant is 0, there is 1 real solution to the equation.

$7q^2-3q-6=0$

The equation is in standard form, identify a, b, and c.

$a=7,b=-3,c=-6$

Write the discriminant.

$b^2-4ac$

Substitute in the values of a, b, and c and simplify.

localid="1663937212334" ${(-3)}^2-4·7·(-6)=9+168=177$

Since the discriminant is positive, there are 2 real solutions to the equation.

$7y^2+2y+8=0$

The equation is in standard form, identify a, b, and c.

$a=7,b=2,c=8$

Write the discriminant.

$b^2-4ac$

Substitute in the values of a, b, and c and simplify.

$2^2-4·7·8=4-224=-220$

Since the discriminant is negative. Thus there are no real solutions to the equation.

The number of solutions for each quadratic equation:

Part (a) 1

Part (b) 2

Part (c) no real solution.