Solve the Quadratic Equation: 3x² + 10x - 8 = 0

Quadratic Formula with Discriminant Analysis

Solve the following equation:

3x2+10x−8=0 3x^2+10x-8=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 Identify the coefficients
00:14 Use the roots formula
00:35 Substitute appropriate values according to the given data and solve
00:59 Calculate the square and products
01:16 Calculate the square root of 196
01:33 These are the 2 possible solutions (addition,subtraction)
01:48 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Solve the following equation:

3x2+10x−8=0 3x^2+10x-8=0

2

Step-by-step solution

To solve the quadratic equation 3x2+10x−8=0 3x^2 + 10x - 8 = 0 , we will apply the quadratic formula.

First, identify the coefficients:
a=3 a = 3 , b=10 b = 10 , c=−8 c = -8 .

Calculate the discriminant Δ \Delta :
Δ=b2−4ac=102−4×3×(−8)=100+96=196 \Delta = b^2 - 4ac = 10^2 - 4 \times 3 \times (-8) = 100 + 96 = 196

Since the discriminant Δ=196 \Delta = 196 is greater than zero, the quadratic equation has two distinct real roots.

Now apply the quadratic formula:
x=−b±Δ2a=−10±1966 x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-10 \pm \sqrt{196}}{6}

Calculate the roots:

  • Root x1 x_1 :
    x1=−10+146=46=23 x_1 = \frac{-10 + 14}{6} = \frac{4}{6} = \frac{2}{3}
  • Root x2 x_2 :
    x2=−10−146=−246=−4 x_2 = \frac{-10 - 14}{6} = \frac{-24}{6} = -4

Thus, the solutions are x1=23 x_1 = \frac{2}{3} and x2=−4 x_2 = -4 .

Therefore, the solution to the equation 3x2+10x−8=0 3x^2 + 10x - 8 = 0 is x1=23,x2=−4 x_1 = \frac{2}{3}, x_2 = -4 .

3

Final Answer

x1=23,x2=−4 x_1=\frac{2}{3},x_2=-4

Key Points to Remember

Essential concepts to master this topic
  • Coefficients: Identify a=3, b=10, c=-8 from standard form
  • Discriminant: Calculate Δ=102−4(3)(−8)=196 \Delta = 10^2 - 4(3)(-8) = 196
  • Verification: Substitute x=23 x = \frac{2}{3} : 3(23)2+10(23)−8=0 3(\frac{2}{3})^2 + 10(\frac{2}{3}) - 8 = 0 ✓

Common Mistakes

Avoid these frequent errors
  • Sign errors when calculating discriminant
    Don't calculate b2−4ac b^2 - 4ac as 102−4(3)(8)=4 10^2 - 4(3)(8) = 4 = wrong discriminant! Forgetting that c = -8 (not +8) changes the entire calculation. Always write out b2−4ac=100−4(3)(−8)=100+96=196 b^2 - 4ac = 100 - 4(3)(-8) = 100 + 96 = 196 .

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

Why do we need the discriminant first?

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The discriminant Δ=b2−4ac \Delta = b^2 - 4ac tells us how many real solutions exist! If Δ>0 \Delta > 0 (like our 196), we get two distinct real roots.

What if I get a negative number under the square root?

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If the discriminant is negative, the quadratic has no real solutions. The parabola doesn't cross the x-axis, so there are no real x-intercepts.

How do I simplify 196 \sqrt{196} ?

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Look for perfect square factors! Since 142=196 14^2 = 196 , we have 196=14 \sqrt{196} = 14 . Practice your perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196...

Can I solve this by factoring instead?

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Yes! Try to find two numbers that multiply to ac=3(−8)=−24 ac = 3(-8) = -24 and add to b=10 b = 10 . Those numbers are 12 and -2, giving us (3x−2)(x+4)=0 (3x - 2)(x + 4) = 0 .

Why are there two different solutions?

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A quadratic equation represents a parabola, and parabolas can cross the x-axis at two different points. Each crossing point gives us a solution to the equation.

How do I check my work?

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Substitute both solutions back into the original equation. For x=23 x = \frac{2}{3} : 3(23)2+10(23)−8=43+203−8=8−8=0 3(\frac{2}{3})^2 + 10(\frac{2}{3}) - 8 = \frac{4}{3} + \frac{20}{3} - 8 = 8 - 8 = 0 ✓

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