Solve Quadratic Equation: -2X² + 6X + 8 = 0

Q: Why is the leading coefficient negative in this problem?

The equation $ -2X^2 + 6X + 8 = 0 $ has a = -2. This just means the parabola opens downward instead of upward, but we solve it the same way using the quadratic formula!

Q: How do I avoid sign mistakes with negative coefficients?

Write out each step clearly! When a = -2, remember that 2a = 2(-2) = -4. Double-check your arithmetic, especially when multiplying negative numbers.

Q: What does the discriminant tell me?

The discriminant $ b^2 - 4ac = 100 $ is positive, so we have two real, distinct solutions. If it were zero, we'd have one solution. If negative, no real solutions.

Q: Why do I get two different answers?

Quadratic equations usually have two solutions because a parabola can cross the x-axis at two points. The ± in the formula gives us both intersection points.

Q: How can I check if X = 4 and X = -1 are correct?

Substitute each value back into the original equation:For X = 4: $ -2(16) + 6(4) + 8 = -32 + 24 + 8 = 0 $ ✓For X = -1: $ -2(1) + 6(-1) + 8 = -2 - 6 + 8 = 0 $ ✓

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Pay attention to the coefficients

00:10 Use the roots formula

00:19 Substitute appropriate values according to the given data and solve for X

00:34 Calculate the multiplications and the square

00:45 Calculate the square root of 100

00:49 Find the two possible solutions

01:02 This is one solution

01:09 And this is the second solution and the answer to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Solve for X:

$-2X^2+6X+8=0$

2

Step-by-step solution

To solve the quadratic equation $-2X^2 + 6X + 8 = 0$ using the quadratic formula, follow these steps:

Step 1: Calculate the discriminant, $b^2 - 4ac$ .

Here, $a = -2$ , $b = 6$ , and $c = 8$ . Plug these into the formula: $b^2 - 4ac = 6^2 - 4(-2)(8) = 36 + 64 = 100$ Since the discriminant is greater than zero, the roots are real and distinct.

Step 2: Apply the quadratic formula, $X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Substituting the values, we have: $X = \frac{-6 \pm \sqrt{100}}{2(-2)}$ Simplifying inside the square root gives us: $X = \frac{-6 \pm 10}{-4}$ This leads to two possible solutions: - First, calculate with the positive square root: $X_1 = \frac{-6 + 10}{-4} = \frac{4}{-4} = -1$ - Second, calculate with the negative square root: $X_2 = \frac{-6 - 10}{-4} = \frac{-16}{-4} = 4$

Thus, the solutions to the equation are $X_1 = 4$ and $X_2 = -1$ .

Verifying against the choices, the correct choice is: : $X_1=4, X_2=-1$ .

Therefore, the solution is $X_1 = 4, X_2 = -1$ .

3

Final Answer

$X_1=4, X_2=-1$

Key Points to Remember

Essential concepts to master this topic

Formula: Use $X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for any quadratic equation
Discriminant: Calculate $b^2 - 4ac = 6^2 - 4(-2)(8) = 100$ first
Verify: Substitute both solutions back: $-2(4)^2 + 6(4) + 8 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Sign errors when calculating with negative leading coefficient
Don't forget that a = -2, so 2a = -4 in the denominator! Students often write +4 instead = wrong signs in final answers. Always double-check your signs when substituting negative coefficients into the quadratic formula.

Practice Quiz

Test your knowledge with interactive questions

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

FAQ

Everything you need to know about this question

Why is the leading coefficient negative in this problem?

+

The equation $-2X^2 + 6X + 8 = 0$ has a = -2. This just means the parabola opens downward instead of upward, but we solve it the same way using the quadratic formula!

How do I avoid sign mistakes with negative coefficients?

+

Write out each step clearly! When a = -2, remember that 2a = 2(-2) = -4. Double-check your arithmetic, especially when multiplying negative numbers.

What does the discriminant tell me?

+

The discriminant $b^2 - 4ac = 100$ is positive, so we have two real, distinct solutions. If it were zero, we'd have one solution. If negative, no real solutions.

Why do I get two different answers?

+

Quadratic equations usually have two solutions because a parabola can cross the x-axis at two points. The ± in the formula gives us both intersection points.

How can I check if X = 4 and X = -1 are correct?

+

Substitute each value back into the original equation:

For X = 4: $-2(16) + 6(4) + 8 = -32 + 24 + 8 = 0$ ✓
For X = -1: $-2(1) + 6(-1) + 8 = -2 - 6 + 8 = 0$ ✓

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