Solve the Quadratic Equation: x²/2 + x - 4 = 0 Step-by-Step

Q: Can I use factoring instead of the quadratic formula?

Yes! After clearing fractions, look for two numbers that multiply to -8 and add to 2. Here: (x + 4)(x - 2) = 0, so x = -4 or x = 2.

Q: How do I check if my discriminant calculation is correct?

Calculate step by step: $ b^2 - 4ac = 2^2 - 4(1)(-8) = 4 + 32 = 36 $. Since 36 > 0, you should get two real solutions.

Quadratic Formula with Fractional Coefficients

Solve the following equation:

$\frac{x^2}{2}+x-4=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Multiply by 2 to eliminate fractions

00:13 Identify the coefficients

00:21 Use the roots formula

00:37 Substitute appropriate values according to the given data and solve

01:05 Calculate the square and products

01:15 Calculate the square root of 36

01:30 These are the 2 possible solutions (addition,subtraction)

01:44 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$\frac{x^2}{2}+x-4=0$

Step-by-step solution

To solve the equation $\frac{x^2}{2} + x - 4 = 0$ , we'll use the quadratic formula. First, we need to rewrite the equation in standard quadratic form, $ax^2 + bx + c = 0$ .

Start by multiplying the entire equation by 2 to eliminate the fraction:

\ x^2 + 2x - 8 = 0

Now, identify the coefficients:

$a = 1$
$b = 2$
$c = -8$

Next, plug these coefficients into the quadratic formula:

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

Substitute $a = 1$ , $b = 2$ , and $c = -8$ into the formula:

\ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1}

Simplify inside the square root first:

\ 2^2 - 4 \cdot 1 \cdot (-8) = 4 + 32 = 36

Substitute back:

\ x = \frac{-2 \pm \sqrt{36}}{2}

The square root of 36 is 6, so:

\ x = \frac{-2 \pm 6}{2}

Calculate the two possible solutions:

x_1 = \frac{-2 + 6}{2} = \frac{4}{2} = 2

x_2 = \frac{-2 - 6}{2} = \frac{-8}{2} = -4

Therefore, the solutions for the equation $\frac{x^2}{2} + x - 4 = 0$ are $x_1 = 2$ and $x_2 = -4$ .

Final Answer

$x_1=2,x_2=-4$

Key Points to Remember

Essential concepts to master this topic

Standard Form: Multiply equation by 2 to eliminate fraction: $x^2 + 2x - 8 = 0$
Quadratic Formula: Use $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with a=1, b=2, c=-8
Verification: Check both solutions: $\frac{2^2}{2} + 2 - 4 = 0$ and $\frac{(-4)^2}{2} + (-4) - 4 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply all terms when clearing fractions
Don't multiply only $\frac{x^2}{2}$ by 2 and leave x - 4 unchanged = wrong equation! This creates an unbalanced equation with incorrect coefficients. Always multiply every single term by 2 to get $x^2 + 2x - 8 = 0$ .

Practice Quiz

Test your knowledge with interactive questions

a = coefficient of x²

b = coefficient of x

c = coefficient of the constant term

What is the value of $$ c $$ in the function $$ y=-x^2+25x $$ ?

FAQ

Everything you need to know about this question

Why do I need to clear the fraction first?

Clearing fractions makes the quadratic formula easier to use! Working with $x^2 + 2x - 8 = 0$ is much simpler than using a = 1/2 in the formula.

What if I forget to multiply all terms by 2?

You'll get the wrong coefficients for the quadratic formula! Always multiply every single term on both sides. Check: $\frac{x^2}{2} \cdot 2 + x \cdot 2 - 4 \cdot 2 = 0 \cdot 2$

Can I use factoring instead of the quadratic formula?

Yes! After clearing fractions, look for two numbers that multiply to -8 and add to 2. Here: (x + 4)(x - 2) = 0, so x = -4 or x = 2.

How do I check if my discriminant calculation is correct?

Calculate step by step: $b^2 - 4ac = 2^2 - 4(1)(-8) = 4 + 32 = 36$ . Since 36 > 0, you should get two real solutions.

What does it mean when I get a perfect square under the radical?

Great news! When the discriminant is a perfect square like 36, your solutions will be rational numbers (no messy radicals). $\sqrt{36} = 6$ exactly.

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