Solve the Quadratic Inequality: x²-3x+4<0 Step by Step

Q: Why does a negative discriminant mean no solution?

When $ \Delta = b^2 - 4ac , the parabola doesn't touch the x-axis at all! Since it opens upward (positive coefficient of \( x^2 $), it stays entirely above the x-axis, always positive.

Q: How do I know if the expression is always positive or negative?

After finding $ \Delta , test any value you want! Pick \( x = 0 $ for simplicity. If positive, the whole expression is always positive. If negative, it's always negative.

Q: What if the question asked for ≥ 0 instead of < 0?

If $ x^2 - 3x + 4 ≥ 0 $, and we know the expression is always positive, then every real number would be a solution! The answer would be $ x \in \mathbb{R} $.

Q: Can I use the quadratic formula here?

You could, but you'd get complex roots like $ x = \frac{3 ± i\sqrt{7}}{2} $. For inequalities, checking the discriminant and testing one point is much faster and clearer!

Q: What does 'no solution' actually mean in math?

It means there are zero values of x that make the inequality true. We can write this as $ x \in \emptyset $ (empty set) or simply state 'no solution'.

Quadratic Inequalities with No Real Solutions

Solve the following equation:

$x^2-3x+4<0$

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Understand the problem

Solve the following equation:

$x^2-3x+4<0$

Step-by-step solution

The problem requires us to solve the inequality $x^2 - 3x + 4 < 0$ .

To solve the inequality, we first consider the corresponding quadratic equation $x^2 - 3x + 4 = 0$ and find its roots.

Calculate the discriminant $\Delta$ :
$\Delta = b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot 4 = 9 - 16 = -7$ .

The discriminant $\Delta = -7$ is less than zero, indicating that the quadratic equation has no real roots. This implies that the quadratic expression $x^2 - 3x + 4$ does not change sign and is either always positive or always negative.

Next, evaluate the sign of $x^2 - 3x + 4$ . For $x = 0$ , the expression is $0^2 - 3 \cdot 0 + 4 = 4$ , which is positive. Therefore, the expression is always positive for all real $x$ .

Since $x^2 - 3x + 4$ is always positive, there is no $x$ for which $x^2 - 3x + 4 < 0$ holds true.

Therefore, the solution to the inequality is that there is no solution, which corresponds to option 4: "There is no solution."

Final Answer

There is no solution.

Key Points to Remember

Essential concepts to master this topic

Discriminant Rule: When Δ < 0, the quadratic has no real roots
Sign Analysis: Test any x-value: $0^2 - 3(0) + 4 = 4 > 0$
Verification: If expression is always positive, inequality < 0 has no solution ✓

Common Mistakes

Avoid these frequent errors

Assuming quadratic inequalities always have solutions
Don't automatically look for interval solutions like (1,4) = wrong answer! When the discriminant is negative, the parabola never crosses the x-axis. Always check the discriminant first and test the sign of the expression.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ x^2+4>0 $$

FAQ

Everything you need to know about this question

Why does a negative discriminant mean no solution?

When $\Delta = b^2 - 4ac < 0$ , the parabola doesn't touch the x-axis at all! Since it opens upward (positive coefficient of $x^2$ ), it stays entirely above the x-axis, always positive.

How do I know if the expression is always positive or negative?

After finding $\Delta < 0$ , test any value you want! Pick $x = 0$ for simplicity. If positive, the whole expression is always positive. If negative, it's always negative.

What if the question asked for ≥ 0 instead of < 0?

If $x^2 - 3x + 4 ≥ 0$ , and we know the expression is always positive, then every real number would be a solution! The answer would be $x \in \mathbb{R}$ .

Can I use the quadratic formula here?

You could, but you'd get complex roots like $x = \frac{3 ± i\sqrt{7}}{2}$ . For inequalities, checking the discriminant and testing one point is much faster and clearer!

What does 'no solution' actually mean in math?

It means there are zero values of x that make the inequality true. We can write this as $x \in \emptyset$ (empty set) or simply state 'no solution'.

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