Solve Quadratic Equation: x² + 3x + 7 = 0 in Standard Form

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

A negative discriminant means the parabola never touches the x-axis. Since \( D = -19 , there are no real number solutions to this equation.

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

Technically yes, but it gives complex solutions involving $ \sqrt{-19} $. Unless specifically asked for complex solutions, the answer is "No solution" in real numbers.

Q: How can I tell before calculating if there might be no solutions?

Look at the equation: $ x^2 + 3x + 7 = 0 $. Since the constant term (7) is large compared to the linear term coefficient (3), the discriminant is likely negative.

Q: Are there other ways to see this has no real solutions?

Yes! You can complete the square: $ x^2 + 3x + 7 = (x + 1.5)^2 + 4.75 = 0 $. Since we need $ (x + 1.5)^2 = -4.75 $, which is impossible for real numbers.

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

Double-check your arithmetic! With $ a=1, b=3, c=7 $: $ D = 3^2 - 4(1)(7) = 9 - 28 = -19 $. The most common error is miscalculating $ 4ac $.

Quadratic Equations with Negative Discriminant

Solve the following equation:

$x^2+3x+7=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 Let's examine the coefficients

00:16 We'll use the root formula

00:34 We'll substitute the appropriate values according to the given data and solve

00:59 Let's calculate the squares and multiplications

01:10 It's impossible for a root to be a negative number

01:14 Therefore, there is no 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:

$x^2+3x+7=0$

Step-by-step solution

To solve the quadratic equation $x^2 + 3x + 7 = 0$ , we will use the quadratic formula:

Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Identify the coefficients: $a = 1$ , $b = 3$ , $c = 7$ .
Calculate the discriminant: $D = b^2 - 4ac = 3^2 - 4(1)(7) = 9 - 28 = -19$ .

The discriminant $D = -19$ is negative. This indicates that there are no real roots to the equation. Quadratic equations with negative discriminants have complex solutions.

Therefore, the given quadratic equation $x^2 + 3x + 7 = 0$ has no solutions in the real number system, and we conclude that:

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
Technique: Calculate $D = 3^2 - 4(1)(7) = 9 - 28 = -19$
Check: Negative discriminant means complex roots, so "No solution" in real numbers ✓

Common Mistakes

Avoid these frequent errors

Continuing with quadratic formula despite negative discriminant
Don't try to find $\sqrt{-19}$ and write complex solutions = wrong answer type! The question asks for real solutions only. Always recognize that negative discriminant means "No solution" in the real number system.

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 never touches the x-axis. Since $D = -19 < 0$ , there are no real number solutions to this equation.

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

Technically yes, but it gives complex solutions involving $\sqrt{-19}$ . Unless specifically asked for complex solutions, the answer is "No solution" in real numbers.

How can I tell before calculating if there might be no solutions?

Look at the equation: $x^2 + 3x + 7 = 0$ . Since the constant term (7) is large compared to the linear term coefficient (3), the discriminant is likely negative.

Are there other ways to see this has no real solutions?

Yes! You can complete the square: $x^2 + 3x + 7 = (x + 1.5)^2 + 4.75 = 0$ . Since we need $(x + 1.5)^2 = -4.75$ , which is impossible for real numbers.

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

Double-check your arithmetic! With $a=1, b=3, c=7$ : $D = 3^2 - 4(1)(7) = 9 - 28 = -19$ . The most common error is miscalculating $4ac$ .

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