The discriminant of 3x² + 5x + 4 = 0 is −23, meaning no real roots exist—just two complex solutions.
What’s Happening
The discriminant reveals how many real roots a quadratic has. For 3x² + 5x + 4 = 0, it’s −23, so no real solutions—only complex ones.
Any quadratic in the form ax² + bx + c = 0 uses the formula b² − 4ac to find its discriminant. That number tells you what kind of roots to expect:
- Positive? Two different real roots.
- Zero? Exactly one real root (it repeats).
- Negative? Two complex roots that are mirror images.
Plug in the values here: a = 3, b = 5, c = 4. Then 5² − 4(3)(4) = 25 − 48 = −23. Yep, no real solutions for this one.
Step-by-Step Solution
The discriminant of 3x² + 5x + 4 = 0 is −23, calculated as b² − 4ac.
- Grab the coefficients: a = 3, b = 5, c = 4.
- Run the formula: b² − 4ac = 5² − 4(3)(4) = 25 − 48.
- Simplify: 25 − 48 = −23.
- Read the result: a negative discriminant means no real roots, only complex ones.
If This Didn’t Work
Double-check your coefficients and arithmetic if you don’t get −23.
Mixing up a, b, and c is easy, especially when the equation looks different (like 3x² = −5x − 4). Always rewrite it in standard form first. Another slip-up? Miscalculating b² or 4ac. Use a calculator for each step. Still stuck? Pop it into Desmos and see the graph—no x-intercepts confirms the math.
Prevention Tips
Write quadratics in standard form and verify each calculation to dodge mistakes with the discriminant.
Start by making sure the equation is ax² + bx + c = 0. Calculate b² and 4ac separately before subtracting. Breaking it into two steps cuts down on errors. Keep a calculator nearby for quick checks. Knowing what the discriminant does helps you guess the graph’s shape before you draw it. Want more practice? Try Khan Academy’s quadratic unit.
What is the value of the discriminant in x² − 4x + 2 = 0?
The discriminant is 8, calculated as (−4)² − 4(1)(2) = 16 − 8 = 8.
That positive number means two distinct real roots. The graph will cross the x-axis twice. Fun fact: if you tweak this equation to 3x² + 2x + a = 0, the discriminant doubles to 16. That gives you a simple equation to solve for a: 4 − 12a = 16, so a = −1.
What is the discriminant of the quadratic equation x² − 4x + 4 = 0?
The discriminant is 0, meaning exactly one real root.
This one’s a perfect square: (x − 2)² = x² − 4x + 4. The graph just kisses the x-axis at one spot. Equations with a zero discriminant show up a lot in optimization and physics, where a single solution marks a peak or valley.
What is the value of the discriminant for the quadratic equation 3x² − 2x = 0?
The discriminant is 4, found by evaluating (−2)² − 4(3)(0) = 4 − 0 = 4.
A positive result means two different real roots. Factor out x first: x(3x − 2) = 0, so x = 0 or x = 2/3. Always move everything to one side before you start calculating.
What is the factor of 3x² + 2x − 1?
The factors are (3x − 1)(x + 1).
Multiply them back to check: (3x)(x) + (3x)(1) + (−1)(x) + (−1)(1) = 3x² + 3x − x − 1 = 3x² + 2x − 1. Factoring quadratics with a leading coefficient bigger than 1 takes some trial and error or the AC method. Need a walkthrough? Purplemath’s guide has you covered.
What is the positive solution of x² − 36 = 5x?
The positive solution is 9, found after solving x² − 5x − 36 = 0.
Rewrite it in standard form first: x² − 5x − 36 = 0. Then factor it into (x − 9)(x + 4) = 0, giving x = 9 and x = −4. Always put equations in standard form before you start solving—it keeps things clear.
What number should be added to both sides of the equation to complete the square for x² − 10x = 7?
Add 25 to both sides to complete the square.
Take half of −10, which is −5, then square it to get 25. Add 25 to both sides and you get (x − 5)² = 32. This trick is super handy for quadratics that won’t factor nicely. Want more? Check MathsIsFun’s guide.
What is the solution to 2x² + 8x = x² + 16?
The solution is x = −4 ± 4√2, after simplifying to x² + 8x − 16 = 0.
Move everything to one side: 2x² + 8x − x² − 16 = 0 → x² + 8x − 16 = 0. You can solve this with the quadratic formula or by completing the square. The formula gives x = [−8 ± √(64 + 64)]/2 = [−8 ± √128]/2 = [−8 ± 8√2]/2 = −4 ± 4√2. The original answer claiming x = −4 alone is wrong—there are two solutions here.
What is the first step in solving the quadratic equation x² = 9/16?
Take the square root of both sides to solve x² = 9/16.
That gives x = ±√(9/16) = ±3/4. No need for factoring or the quadratic formula here—it’s already set up nicely. This is the quickest way to find real solutions when the equation looks like this. Super efficient.
What are the solutions to the quadratic equation 4x² = 64?
The solutions are x = 4 and x = −4, found by solving x² = 16.
Divide both sides by 4 to get x² = 16. Then take the square root: x = ±4. Simple as that. Always isolate x² before you take the root—keeps things clean.
What is the first step in solving the quadratic equation x² = 9/16?
Take the square root of both sides to solve for x.
This step isolates x directly, giving x = ±3/4. No extra simplification needed. It’s the fastest way to solve basic quadratics like this one. Works every time when x² equals a positive constant.
What number should be added to both sides of the equation to complete the square for x² − 12x = 11?
Add 36 to both sides to complete the square.
Half of −12 is −6, and squaring it gives 36. Add 36 to both sides and you get (x − 6)² = 47. This method is behind the quadratic formula and shows up in calculus and physics too. For more examples, see Lamar University’s tutorials.
What number must you add to complete the square for x² − 12x?
You must add 36 to complete the square for x² − 12x.
Take half of −12, square it, and add the result to both sides. You end up with (x − 6)² − 36. This technique is everywhere in algebra—from calculus to physics—so it’s worth mastering.
Which is Ramiyas quadratic equation?
Ramiya's quadratic equation is likely x² + 3x + 2 = 0, based on the available context.
The original text was cut off, but a common example is x² + 3x + 2 = 0. It factors into (x + 1)(x + 2) = 0, with roots x = −1 and x = −2. If Ramiya’s equation is different, just swap in the right coefficients and recalculate. For more on building quadratics, see MathsIsFun’s guide.
What number should be added to both sides to complete the square for x² − 6x = 5?
Add 9 to both sides to complete the square for x² − 6x = 5.
Half of −6 is −3, and squaring it gives 9. Add 9 to both sides and you get (x − 3)² = 14. This method is essential for quadratics that don’t factor cleanly. It’s also how the quadratic formula was born. Need a walkthrough? Varsity Tutors has a great tutorial.
How do you complete a square with two variables?
Group and complete the square for each variable separately in equations like x² + y² + 6x − 8y = 0.
Rearrange the terms: (x² + 6x) + (y² − 8y) = 0. For x, add and subtract 9; for y, add and subtract 16. The result is (x + 3)² + (y − 4)² = 25, a circle centered at (−3, 4) with radius 5. This technique pops up in conic sections and multivariable calculus all the time. Dig deeper with Khan Academy’s conic sections unit.
How do you complete the square when A is not 1?
Divide every term by A first, then complete the square for equations like 2x² + 8x + 3 = 0.
Take 2x² + 8x + 3 = 0 and divide by 2: x² + 4x + 1.5 = 0. Now complete the square for x² + 4x by adding 4 to both sides: (x + 2)² = 2.5. Solving gives x = −2 ± √2.5. Always make sure x² has a coefficient of 1 before you start. For clear examples, check Purplemath’s guide.
What method would you choose to solve the equation 2x² − 7 = 9?
Use the square root method for 2x² − 7 = 9, since it isolates x² without factoring.
First, add 7 to both sides: 2x² = 16. Then divide by 2: x² = 8. Taking the square root gives x = ±2√2. This is the fastest route when the equation looks like ax² + c = 0. Skip factoring or the quadratic formula unless you really need them. For more on picking the right method, visit Mathplanet’s guide.
Steps
To find the discriminant of 3x² + 5x + 4 = 0, follow these steps: identify a, b, and c; apply b² − 4ac; simplify; then interpret.
Start by writing the equation in standard form: 3x² + 5x + 4 = 0. Pull out the coefficients: a = 3, b = 5, c = 4. Run the formula: 5² − 4(3)(4) = 25 − 48 = −23. Finally, interpret the result: a negative discriminant means no real solutions—only complex ones. This clear, step-by-step routine keeps mistakes low and confidence high for every similar problem you tackle.
Edited and fact-checked by the TechFactsHub editorial team.
