Solve the Quadratic Equation: x²-2x+8=0 Step-by-Step

Q: What does it mean when the discriminant is negative?

A negative discriminant means the parabola doesn't touch or cross the x-axis. The equation has no real solutions, only complex (imaginary) ones involving $ i = \sqrt{-1} $.

Q: How can I tell without calculating if there are no real solutions?

Look at the graph! If the parabola opens upward (positive a) and its vertex is above the x-axis, or opens downward (negative a) with vertex below the x-axis, there are no real solutions.

Q: Did I make an error in my calculation if I get no solutions?

No! Many quadratic equations have no real solutions - this is completely normal. Double-check your discriminant calculation: $ \Delta = b^2 - 4ac $. If it's negative, you're correct!

Q: What's the difference between no solution and no real solution?

'No solution' means the equation cannot be solved at all. 'No real solution' means there are complex solutions involving imaginary numbers, but no solutions on the real number line.

Q: Can I still use other methods like factoring?

You can try, but factoring won't work for quadratics with no real solutions. The discriminant method is the most reliable way to identify when an equation has no real solutions.

Quadratic Equations with Negative Discriminant

Solve the following equation:

$x^2-2x+8=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's find the value of X.

00:09 First, let's identify the coefficients. Pay attention to the numbers in front of the variables.

00:21 Next, we'll use the roots formula. This helps us find solutions to the equation.

00:50 Now, substitute the correct values based on the data we have, and let's solve the equation.

01:18 Let's calculate the square and the products step by step.

01:29 Remember! A root cannot be negative. Keep this in mind.

01:38 So, there's no solution to this question. Great try, and keep practicing!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$x^2-2x+8=0$

Step-by-step solution

To solve the quadratic equation $x^2 - 2x + 8 = 0$ , we'll apply the quadratic formula method. The process is as follows:

Step 1: Identify the coefficients from the equation: $a = 1$ , $b = -2$ , $c = 8$ .
Step 2: Calculate the discriminant using the formula $\Delta = b^2 - 4ac$ .
Step 3: For the given equation, calculate $\Delta = (-2)^2 - 4(1)(8) = 4 - 32 = -28$ .
Step 4: Interpret the discriminant. Since $\Delta = -28$ is negative, this indicates there are no real solutions to the equation.

Without real roots, the equation has complex solutions. Since the problem requires real solutions, we conclude:

The equation $x^2 - 2x + 8 = 0$ has no real solution.

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic

Discriminant Rule: When $\Delta = b^2 - 4ac < 0$ , no real solutions exist
Calculation: For $x^2 - 2x + 8 = 0$ , discriminant is $(-2)^2 - 4(1)(8) = -28$
Verification: Negative discriminant means parabola never crosses x-axis ✓

Common Mistakes

Avoid these frequent errors

Ignoring the negative discriminant and proceeding with quadratic formula
Don't continue calculating $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ when discriminant is negative = impossible square root of negative number! This creates mathematical errors and confusion. Always check the discriminant first and conclude 'no real solutions' when $\Delta < 0$ .

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 the discriminant is negative?

A negative discriminant means the parabola doesn't touch or cross the x-axis. The equation has no real solutions, only complex (imaginary) ones involving $i = \sqrt{-1}$ .

How can I tell without calculating if there are no real solutions?

Look at the graph! If the parabola opens upward (positive a) and its vertex is above the x-axis, or opens downward (negative a) with vertex below the x-axis, there are no real solutions.

Did I make an error in my calculation if I get no solutions?

No! Many quadratic equations have no real solutions - this is completely normal. Double-check your discriminant calculation: $\Delta = b^2 - 4ac$ . If it's negative, you're correct!

What's the difference between no solution and no real solution?

'No solution' means the equation cannot be solved at all. 'No real solution' means there are complex solutions involving imaginary numbers, but no solutions on the real number line.

Can I still use other methods like factoring?

You can try, but factoring won't work for quadratics with no real solutions. The discriminant method is the most reliable way to identify when an equation has no real solutions.

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