Solve the Quadratic Equation: Discover x in 7x² + 3x + 8 = 9x + 3

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

When \( \Delta = b^2 - 4ac , the quadratic equation has no real solutions. This means the parabola doesn't cross the x-axis at any point!

Q: Can I still use the quadratic formula with a negative discriminant?

You could use it, but you'd get complex numbers (involving √-104). For most algebra problems, we only want real solutions, so "No solution" is the correct answer.

Q: How do I know I did the algebra correctly when rearranging?

Double-check your work! From $ 7x^2 + 3x + 8 = 9x + 3 $, subtract 9x and 3 from both sides: $ 7x^2 - 6x + 5 = 0 $.

Q: What if I made an error and got a positive discriminant?

Go back and carefully re-check your algebra! Common errors include sign mistakes when moving terms or combining like terms. The correct form should be $ 7x^2 - 6x + 5 = 0 $.

Q: Are there any quadratic equations that truly have no solutions?

Yes! When the discriminant is negative, the quadratic has no real solutions. This happens when the parabola opens upward but never touches the x-axis, or opens downward but stays below the x-axis.

Quadratic Equations with Negative Discriminant

Given the following equation, find its solution

$7x^2+3x+8=9x+3$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:06 Arrange the equation so that the right side equals 0

00:09 Combine like terms

00:18 Identify the coefficients

00:23 Use the quadratic formula to find possible solutions

00:34 Substitute appropriate values and solve to find solutions

00:44 Calculate the square and products

00:52 The root expression is less than 0, therefore no solution exists

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the following equation, find its solution

$7x^2+3x+8=9x+3$

Step-by-step solution

To solve the equation $7x^2 + 3x + 8 = 9x + 3$ , follow these steps:

Step 1: Rearrange the equation into standard quadratic form:
Move all terms to one side:
$7x^2 + 3x + 8 - 9x - 3 = 0$ .
Step 2: Simplify the equation:
Combine like terms:
$7x^2 - 6x + 5 = 0$ .
Step 3: Identify coefficients:
$a = 7$ , $b = -6$ , and $c = 5$ .
Step 4: Calculate the discriminant ( $\Delta$ ):
$\Delta = b^2 - 4ac = (-6)^2 - 4(7)(5) = 36 - 140 = -104$ .
Step 5: Determine the nature of the roots:
Since the discriminant is negative ( $\Delta = -104$ ), this means there are no real solutions.

Therefore, the solution to the equation is No solution.

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic

Standard Form: Always rearrange to ax² + bx + c = 0 first
Discriminant Check: Calculate b² - 4ac = (-6)² - 4(7)(5) = -104
Verification: Negative discriminant means no real solutions exist ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check the discriminant before using the quadratic formula
Don't jump straight to the quadratic formula without checking b² - 4ac first = wasting time on impossible calculations! A negative discriminant tells you immediately there are no real solutions. Always calculate the discriminant first to determine if real solutions exist.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ 2x^2-10x-12=0 $$

FAQ

Everything you need to know about this question

What does it mean when the discriminant is negative?

When $\Delta = b^2 - 4ac < 0$ , the quadratic equation has no real solutions. This means the parabola doesn't cross the x-axis at any point!

Can I still use the quadratic formula with a negative discriminant?

You could use it, but you'd get complex numbers (involving √-104). For most algebra problems, we only want real solutions, so "No solution" is the correct answer.

How do I know I did the algebra correctly when rearranging?

Double-check your work! From $7x^2 + 3x + 8 = 9x + 3$ , subtract 9x and 3 from both sides: $7x^2 - 6x + 5 = 0$ .

What if I made an error and got a positive discriminant?

Go back and carefully re-check your algebra! Common errors include sign mistakes when moving terms or combining like terms. The correct form should be $7x^2 - 6x + 5 = 0$ .

Are there any quadratic equations that truly have no solutions?

Yes! When the discriminant is negative, the quadratic has no real solutions. This happens when the parabola opens upward but never touches the x-axis, or opens downward but stays below the x-axis.

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