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

$$\begin{gathered} \left(a\right)5x^2-7x-8=0 \\ \left(b\right)7x^2-10x+5=0 \\ \left(c\right)25x^2-90x+81=0 \\ \left(d\right)15x^2-8x+4=0 \end{gathered}$$

$5x^2-7x-8=0$

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

$a=5,b=-7,c=-8$

Write in discriminant, and substitute the values ofa, b,andc:

$$\begin{gathered} b^2=4ac \\ ={\left(-7\right)}^2-4\left(5\right)\left(-8\right) \\ =49+160 \\ =209 \end{gathered}$$

As the discriminant is positive, there will be two real solutions for this equation.

The given equation is:

$7x^2-10x+5=0$

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

$a=7,b=-10,c=5$

Write in discriminant, and substitute the values ofa, b,andc:

$$\begin{gathered} b^2-4ac \\ ={\left(-10\right)}^2-4\left(7\right)\left(5\right) \\ =100-140 \\ =-40 \end{gathered}$$

As the discriminant is negative, there will be no real solution for this equation.

$25x^2-90x+81=0$

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

$a=25,b=-90,c=81$

Write in discriminant, and substitute the values ofa, b,andc:

$$\begin{gathered} b^2-4ac \\ ={\left(-90\right)}^2-4\left(25\right)\left(81\right) \\ =8100-8100 \\ =0 \end{gathered}$$

As the discriminant is zero, there will be only one real solution for this equation.

$15x^2-8x+4=0$

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

$a=15,b=-8,c=4$

Write in discriminant, and substitute the values ofa, b,andc:

$$\begin{gathered} b^2-4ac \\ ={\left(-8\right)}^2-4\left(15\right)\left(4\right) \\ =64- \\ =-176 \end{gathered}$$

As the discriminant is negative, there will be no real solution for this equation.