Question

For the following equation: 2x^2 -3x + 7 = 0
(1) Calculate the discriminant
(2) Determine the number and type of solutions
(3) Use the quadratic formula to solve

Answer 1

Answer:

Part 1)

$\Delta=-47$

Part 2)

Since our discriminant is negative, we will have two complex (imaginary) solutions.

Part 3)

$\displaystyle x_1=\frac{3}{4}+\frac{\sqrt{47}}{4}i\text{ and } x_2=\frac{3}{4}-\frac{\sqrt{47}}{4}i$

Step-by-step explanation:

We have the equation:

$2x^2-3x+7=0$

Labelling our coefficients, we see that:

$a=2, b=-3, \text{ and } c=7$

Part 1)

The discriminant (symbolized as Δ) is given by the formula:

$\displaystyle \Delta=b^2-4ac$

So, the value of our discriminant is:

$\displaystyle \Delta&=(-3)^2-4(2)(7)$

Evaluate:

$\Delta=9-56=-47$

Part 2)

Remember the guidelines for the discriminant:

• If Δ>0 (the discriminant is positive), then our equation has two real solutions.
• If Δ<0 (the discriminant is negative), then our equation has two complex solutions (imaginary).
• If Δ=0 (the discriminant is 0), then our equation has exactly one real root.

Since our discriminant is a negative value, we will have two complex (imaginary) roots.

Part C)

The quadratic formula is given by:

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

By substitution:

$\displaystyle x=\frac{-(-3)\pm\sqrt{(-3)^2-4(2)(7)}}{2(2)}$

Evaluate:

$\displaystyle x=\frac{3\pm\sqrt{-47}}{4}$

Simplify the square root:

$\sqrt{-47}=\sqrt{-1\cdot47}=i\sqrt{47}$

Therefore:

$\displaystyle x=\frac{3\pm i\sqrt{47}}{4}$

Hence, our two zeros are:

$\displaystyle x_1=\frac{3+i\sqrt{47}}{4}\text{ and } x_2=\frac{3-i\sqrt{47}}{4}$

And in standard form:

$\displaystyle x_1=\frac{3}{4}+\frac{\sqrt{47}}{4}i\text{ and } x_2=\frac{3}{4}-\frac{\sqrt{47}}{4}i$

Answer 2

Answer:

the discriminant is b squared - 4 (a) (c) where a = 2, b = -3, and c = 7

after plugging in, the discrimimant is -47

if the discriminant is negative then it has no 'real' roots (the graph will not intersect the x-axis) but it has 2 solutions:  (3+47i)/2 and (3-47i)/2 where 47i equals the square root of negative 47

Step-by-step explanation: