Quick Fix Summary
What’s Happening
Ever wonder why some quadratic equations have real solutions while others don’t? That’s where the discriminant comes in. For any quadratic in the form ax² + bx + c = 0, the discriminant is b² − 4ac. This little number tells us everything about the roots:
- Positive? You get two distinct real roots.
- Zero? Exactly one real root (a repeated one).
- Negative? No real roots—just complex ones. (Honestly, this is the trickiest case for beginners.)
Plugging in our equation, 3x² + 5x + 4 = 0, gives us (5)² − 4(3)(4) = 25 − 48 = −23. The negative result? No real solutions for this one.
Step-by-Step Solution
- Grab the coefficients: For 3x² + 5x + 4 = 0, a = 3, b = 5, and c = 4. Write them down clearly—don’t mix these up!
- Plug into the formula: Now substitute those values into b² − 4ac:
- 5² − 4(3)(4)
- 25 − 48
- −23
- What does it mean? A discriminant of −23 confirms no real roots. If you need the complex ones, the quadratic formula will handle it. (That’s a story for another day.)
If This Didn’t Work
Made a mistake? Happens to everyone. Here’s how to troubleshoot:
- Check your coefficients: Did you mix up a, b, or c? For example, 3x² + 5x − 4 = 0 would give a totally different discriminant.
- Standard form matters: Equations must look like ax² + bx + c = 0. If yours is scrambled (like 3x² = −5x − 4), rearrange it first.
- Calculator to the rescue: Tools like WolframAlpha or a graphing calculator can double-check your work. Just input the equation and compare results.
Prevention Tips
Want to avoid mistakes next time? Follow these habits:
- Start standardized: Always write quadratics as ax² + bx + c = 0 before calculating.
- Double-check math: Calculate b² and 4ac separately, then subtract. Two passes beat one every time.
- Know the takeaway: A negative discriminant? No real roots. Useful for graphing or real-world problems. For deeper dives, check out the Khan Academy quadratic guide.
What is the value of the discriminant in x² − 4x + 2 = 0?
Now, let’s find the discriminant of x² − 4x + 2 = 0. The formula b² − 4ac gives us (−4)² − 4(1)(2) = 16 − 8 = 8. So the discriminant is 8. Fun fact: The problem also mentions another equation, 3x² + 2x + a = 0, where its discriminant is double this value. That sets up a simple linear equation for a.
What is the discriminant of the quadratic equation x² − 4x + 4 = 0?
Zero. That’s why this equation has exactly one real root (a repeated solution).
What is the value of the discriminant for the quadratic equation 3x² − 2x = 0?
16. Here, a = 3, b = −2, and c = 0. Plugging in: (−2)² − 4(3)(0) = 4 − 0 = 4. Wait—why did we say 16 earlier? That was a typo. The correct discriminant is 4.
What is the factor of 3x² + 2x − 1?
(3x − 1)(x + 1). Multiply it back to check: (3x)(x) + (3x)(1) + (−1)(x) + (−1)(1) = 3x² + 3x − x − 1 = 3x² + 2x − 1. Perfect!
What is the positive solution of x² − 36 = 5x?
9. First, rewrite it as x² − 5x − 36 = 0. Then use the quadratic formula or factor it: (x − 9)(x + 4) = 0. The solutions are 9 and −4, so the positive one is 9.
What number should be added to both sides of the equation to complete the square for x² − 10x = 7?
25. Here’s why: Take half of −10 (which is −5), square it (25), and add it to both sides. Now you have (x − 5)² = 32, ready to solve.
What is the solution to 2x² + 8x = x² + 16?
x = −4. First, simplify: 2x² + 8x − x² − 16 = 0 → x² + 8x − 16 = 0. Then factor or use the quadratic formula. Either way, x = −4 is the only solution.
What is the first step in solving the quadratic equation x² = 9/16?
Take the square root of both sides. That’s it. No need for extra steps here—just x = ±3/4.
What are the solutions to the quadratic equation 4x² = 64?
x = 4 and x = −4. Divide both sides by 4 first: x² = 16. Then take the square root. Simple!
What is the first step in solving the quadratic equation x² = 9/16?
Take the square root of both sides. (Yes, this is the same as the earlier question—but it’s worth repeating.) No complicated steps needed here.
What number should be added to both sides of the equation to complete the square for x² − 12x = 11?
36. Half of −12 is −6, and squaring it gives 36. Add that to both sides to get (x − 6)² = 47.
What number must you add to complete the square for x² − 12x?
36. Same logic as above—half of −12 is −6, squared is 36. Add it to both sides to complete the square.
Which is Ramiyas quadratic equation?
Ramiya’s equation is x = [−3 ± √(3² − 4(1)(2))]/2(1). She substituted a = 1, b = 3, and c = 2 into the quadratic formula. The result? Two possible solutions depending on the ±.
What number should be added to both sides to complete the square for x² − 6x = 5?
9. Half of −6 is −3, and squaring it gives 9. Add 9 to both sides to form (x − 3)² = 14.
How do you complete a square with two variables?
Completing the square with two variables? Start by writing the quadratic in standard form. If the leading coefficient isn’t 1, factor it out of the first two terms. Then take half the coefficient of x, square it, and add it inside the parentheses. (Yes, it’s a bit more work—but doable!)
How do you complete the square when A is not 1?
When A ≠ 1, factor it out from the x² and x terms first. For example, in 2x² − 7x + 3, factor out 2: 2(x² − (7/2)x) + 3. Then complete the square inside the parentheses. It’s all about keeping things balanced.
What method would you choose to solve the equation 2x² − 7 = 9?
I’d isolate the x² term first. Add 7 to both sides: 2x² = 16. Then divide by 2: x² = 8. Finally, take the square root. This method is straightforward—no need for factoring or the quadratic formula here.
Steps
No steps provided—just a placeholder heading. If you’re solving a quadratic, start by identifying the method (factoring, quadratic formula, or completing the square) based on the equation’s form.
