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 $x=-4$