$$ 2 x^{2}=-x+5( $$ Are the roots real or Unreal

Question

UpStudy AI · Accepted Answer

Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$2x^{2}+x-5=0$$
- step1: Solve using the quadratic formula:
  $$x=\frac{-1\pm \sqrt{1^{2}-4\times 2\left(-5\right)}}{2\times 2}$$
- step2: Simplify the expression:
  $$x=\frac{-1\pm \sqrt{1^{2}-4\times 2\left(-5\right)}}{4}$$
- step3: Simplify the expression:
  $$x=\frac{-1\pm \sqrt{41}}{4}$$
- step4: Separate into possible cases:
  $$\begin{align}&x=\frac{-1+\sqrt{41}}{4}\&x=\frac{-1-\sqrt{41}}{4}\end{align}$$
- step5: Rewrite the fraction:
  $$\begin{align}&x=\frac{-1+\sqrt{41}}{4}\&x=-\frac{1+\sqrt{41}}{4}\end{align}$$
- step6: Rewrite:
  $$x_{1}=-\frac{1+\sqrt{41}}{4},x_{2}=\frac{-1+\sqrt{41}}{4}$$
To determine whether the roots of the equation $$ 2x^{2} = -x + 5 $$ are real or unreal, we first rearranged the equation into standard form:

$$
2x^{2} + x - 5 = 0
$$

Next, we solved this quadratic equation and found the roots:

$$
x_{1} = -\frac{1 + \sqrt{41}}{4}, \quad x_{2} = \frac{-1 + \sqrt{41}}{4}
$$

Since the roots involve the square root of a positive number ($$\sqrt{41}$$), both roots are real numbers. Therefore, the roots of the equation are real.
The roots of the equation $$ 2x^{2} = -x + 5 $$ are real.

Are the roots real or Unreal

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