7) Find the discriminant to determine the number of roots, then solve the quadratic equation using the quadratic formula.

Question

2x^2+7x-4=0
____ discriminant
____ number of roots
____ solution

a=-81
b=2
c=1/2 and -4
d= -1/2 and 4
e= 81
f= 0
g= no solution
h=-2
i= 1

Answer 1

To solve the quadratic equation \(2x^2 + 7x - 4 = 0\), we first need to find the discriminant, which is given by the formula:

\[ D = b^2 - 4ac \]

In this equation, \(a = 2\), \(b = 7\), and \(c = -4\).

Let's calculate the discriminant:

\[ D = 7^2 - 4 \cdot 2 \cdot (-4) \] \[ D = 49 + 32 \] \[ D = 81 \]

Discriminant:

• The discriminant \(D = 81\).

Number of Roots:

The number of roots can be determined from the value of the discriminant:

• If \(D > 0\), there are 2 distinct real roots.
• If \(D = 0\), there is exactly 1 real root.
• If \(D < 0\), there are no real roots.

Since \(D = 81\) (which is greater than 0), there are 2 distinct real roots.

Solutions:

Now we will use the quadratic formula to find the roots of the equation:

\[ x = \frac{-b \pm \sqrt{D}}{2a} \]

Using the values \(b = 7\), \(D = 81\), and \(a = 2\):

\[ x = \frac{-7 \pm \sqrt{81}}{2 \cdot 2} \] \[ x = \frac{-7 \pm 9}{4} \]

Calculating the two possible values for \(x\):

1. For \(x = \frac{-7 + 9}{4}\): \[ x = \frac{2}{4} = \frac{1}{2} \]

2. For \(x = \frac{-7 - 9}{4}\): \[ x = \frac{-16}{4} = -4 \]

Solutions:

• The solutions are \(x = \frac{1}{2}\) and \(x = -4\).

Summary:

• Discriminant: 81
• Number of Roots: 2 distinct real roots
• Solutions: \(x = \frac{1}{2}\) and \(x = -4\)