Question

Determine the number and type of solutions to each quadratic equation.

$$\begin{gathered} a)b^2+7b-13=0 \\ b)5a^2-6a+10=0 \\ c)4r^2-20r+25=0 \end{gathered}$$

Short answer

The number and type of solution are

a) as discriminant is greater than zero the equation has 2 real solution

b) as discriminant is less than zero the equation has 2 complex solution

c) as discriminant is equal to zero the equation has 1 real solution.

Step-by-step solution

Part a) Step 1: Given information

We are given a quadratic equation$b^2+7b-13=0$

Part a) step 2: Identify the values of a,b,c

On comparing with standard equation we get$a=1,b=7,c=-13$

Part a) Step 3: Find the value of discriminant

We have,

$$\begin{gathered} =b^2-4ac \\ =49-4\left(1\right)(-13) \\ =49+52 \\ =101 \end{gathered}$$

As the values of discriminant is greater than zero the equation has 2 real solution

Part b) Step 1: Given information

We are given a quadratic equation$5a^2-6a+10=0$

Part b) Step 2: Identify the values of a,b,c

On comparing with standard equation we get$a=5,b=-6,c=10$

Part b) Step 3: Find the value of discriminant

We have

$$\begin{gathered} =b^2-4ac \\ =36-4\left(5\right)\left(10\right) \\ =36-200 \\ =-164 \end{gathered}$$

As the value of discriminant is less than zero the equation has 2 complex roots

Part c) Step 1: Given information

We are given a quadratic equation$4r^2-20r+25=0$

Part c) Step 2: Identify the values of a,b,c

On comparing with standard equation we get$a=4,b=-20,c=25$

Part c) Step 3: Find the value of discriminant 

We have,

$$\begin{gathered} =b^2-4ac \\ =400-4\left(4\right)(25 \\ =400-400 \\ =0 \end{gathered}$$

as the value of discriminant is zero the equation has 1 real root

Conclusion

The number and type of solution are

a) as discriminant is greater than zero the equation has 2 real solution

b) as discriminant is less than zero the equation has 2 complex solution

c) as discriminant is equal to zero the equation has 1 real solution.