Solve the Quadratic Equation: -6x² + 12x - 14 = 0

Q: What does it mean when there are no real solutions?

It means the parabola doesn't cross the x-axis at any point! Since this equation has \( a = -6 , the parabola opens downward but stays entirely below the x-axis.

Q: Why can't I take the square root of -192?

In real numbers, we can't take square roots of negative values. The square root of -192 would involve imaginary numbers, which are beyond basic algebra.

Q: How do I know the discriminant is negative before calculating?

Always calculate $ b^2 - 4ac $ step by step: $ 12^2 - 4(-6)(-14) = 144 - 336 $. The key is being careful with negative signs!

Q: Could I have solved this by factoring instead?

You could try, but since there are no real solutions, the quadratic won't factor using real numbers. The quadratic formula with discriminant checking is the most reliable method.

Q: What if I made a sign error in my discriminant calculation?

Double-check each step: $ b^2 = 144 $, $ 4ac = 4(-6)(-14) = 336 $, so $ 144 - 336 = -192 $. Sign errors are common with negative coefficients!

Quadratic Equations with Negative Discriminants

Solve the following equation:

$-6x^2+12x-14=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Pay attention to coefficients

00:10 Use the roots formula

00:27 Substitute appropriate values according to the given data and solve for X

00:48 Calculate the products and square

01:05 There's no such thing as a root of a negative number, therefore there's no solution

01:10 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:

$-6x^2+12x-14=0$

Step-by-step solution

To solve the quadratic equation $-6x^2 + 12x - 14 = 0$ , we'll use the quadratic formula:

The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Let's calculate step by step:

Identify the coefficients: Here, $a = -6$ , $b = 12$ , and $c = -14$ .
Compute the discriminant: $b^2 - 4ac = 12^2 - 4(-6)(-14) = 144 - 336 = -192$ .
Assess the discriminant: The discriminant is $-192$ , which is less than zero.
Since the discriminant is negative, there are no real solutions to the equation.

Therefore, the correct answer to the problem is "No solution."

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic

Discriminant Rule: When $b^2 - 4ac < 0$ , no real solutions exist
Formula: $b^2 - 4ac = 144 - 336 = -192$ indicates no real roots
Check: Negative discriminant always means no real solutions ✓

Common Mistakes

Avoid these frequent errors

Ignoring the negative discriminant and continuing calculations
Don't keep applying the quadratic formula when the discriminant is negative = impossible square root of negative number! This leads to confusion with imaginary numbers. Always check the discriminant first and conclude 'no real solutions' when it's negative.

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

What does it mean when there are no real solutions?

It means the parabola doesn't cross the x-axis at any point! Since this equation has $a = -6 < 0$ , the parabola opens downward but stays entirely below the x-axis.

Why can't I take the square root of -192?

In real numbers, we can't take square roots of negative values. The square root of -192 would involve imaginary numbers, which are beyond basic algebra.

How do I know the discriminant is negative before calculating?

Always calculate $b^2 - 4ac$ step by step: $12^2 - 4(-6)(-14) = 144 - 336$ . The key is being careful with negative signs!

Could I have solved this by factoring instead?

You could try, but since there are no real solutions, the quadratic won't factor using real numbers. The quadratic formula with discriminant checking is the most reliable method.

What if I made a sign error in my discriminant calculation?

Double-check each step: $b^2 = 144$ , $4ac = 4(-6)(-14) = 336$ , so $144 - 336 = -192$ . Sign errors are common with negative coefficients!

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