Question

Quadratic Equation: -4x^2 = x - 1 Use the discriminant, b^2 - 4ac, to determine the number of solutions of the given quadratic equation. Then solve the quadratic equation using the FORMULA x =FORMULA is in the pic attached

Answer 1

Given the equation:

`-4x^2=\text{ x - 1}`

If we re-arrange the equation

`4x^2\text{ + x - 1 = 0}`

To determine the number of solutions,

Step 1: compare with the equation

`ax^2\text{ + bx + c}`

So that a = 4

b = 1

c = -1

Step 2: Substitute the values of a, b, and c into the discriminant

`D\text{= }b^2\text{ - 4ac}`

Where D is the discriminant

`\begin{gathered} D=1^2\text{ - 4 x 4 x -1} \\ D\text{ = 1 -(-16)} \\ D=\text{ 1+ 16} \\ D=\text{ 17} \end{gathered}`

Since the discriminant, D, is greater than 1, It has two real roots.

Hence, the number of solutions is Two different real solutions

For the second part of the question

`x\text{ = }\frac{-b\text{ }\pm\sqrt[]{b^2\text{ -4ac}}}{2a}`
`D\text{ = }b^2\text{ -4ac = 17}`

`x\text{ =}\frac{-1\pm\sqrt[]{17}\text{ }}{2\text{ x 4}}`
`x\text{ =}\frac{-1\text{ }\pm\sqrt[]{17}}{8}`

Therefore,

`x\text{ =}\frac{-1\text{ +}\sqrt[]{17}}{8}\text{ , x =}\frac{-1\text{ -}\sqrt[]{17}}{8}`