Solve the Quadratic Equation: 4x² - 6x - 4 = 0 Step-by-Step

Quadratic Formula with Integer Discriminant

Solve the following equation:

4x26x4=0 4x^2-6x-4=0

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find X
00:03 Minimize as much as possible
00:19 Identify the coefficients
00:30 Use the roots formula
00:47 Substitute appropriate values according to the given data and solve
01:18 Calculate the squares and products
01:39 Calculate the square root of 25
01:47 These are the 2 possible solutions (addition, subtraction)
02:13 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:

4x26x4=0 4x^2-6x-4=0

2

Step-by-step solution

To solve the quadratic equation 4x26x4=0 4x^2 - 6x - 4 = 0 , we will apply the quadratic formula.

Given the quadratic equation is of the form ax2+bx+c=0 ax^2 + bx + c = 0 , we identify:

  • a=4 a = 4
  • b=6 b = -6
  • c=4 c = -4

Next, we use the quadratic formula:

x=b±b24ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Calculate the discriminant:

b24ac=(6)24×4×(4) b^2 - 4ac = (-6)^2 - 4 \times 4 \times (-4)

b24ac=36+64=100 b^2 - 4ac = 36 + 64 = 100

Since the discriminant is positive, we have two real solutions.

Now, plug the values into the quadratic formula:

x=(6)±1002×4 x = \frac{-(-6) \pm \sqrt{100}}{2 \times 4}

x=6±108 x = \frac{6 \pm 10}{8}

Solving for the two values of x x :

  • First solution: x1=6+108=168=2 x_1 = \frac{6 + 10}{8} = \frac{16}{8} = 2
  • Second solution: x2=6108=48=12 x_2 = \frac{6 - 10}{8} = \frac{-4}{8} = -\frac{1}{2}

Therefore, the solutions to the equation are x1=2 x_1 = 2 and x2=12 x_2 = -\frac{1}{2} .

3

Final Answer

x1=2,x2=12 x_1=2,x_2=-\frac{1}{2}

Key Points to Remember

Essential concepts to master this topic
  • Coefficients: Identify a = 4, b = -6, c = -4 from standard form
  • Discriminant: Calculate b24ac=36+64=100 b^2 - 4ac = 36 + 64 = 100
  • Verification: Check both solutions: 4(2)26(2)4=0 4(2)^2 - 6(2) - 4 = 0

Common Mistakes

Avoid these frequent errors
  • Using wrong signs in quadratic formula
    Don't use -b as -(-6) = -6! This gives x=6±108 x = \frac{-6 ± 10}{8} instead of the correct x=6±108 x = \frac{6 ± 10}{8} . Always remember -b means take the opposite of b, so -(-6) = +6.

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 is the discriminant important?

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The discriminant b24ac b^2 - 4ac tells you how many solutions exist! If it's positive (like 100 here), you get two real solutions. If zero, one solution. If negative, no real solutions.

What if I can't factor this quadratic?

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That's exactly when the quadratic formula is your best friend! Some quadratics like 4x26x4=0 4x^2 - 6x - 4 = 0 don't factor nicely, but the formula always works.

How do I remember the quadratic formula?

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Try this song: "x equals negative b, plus or minus the square root, of b squared minus 4ac, all over 2a!" Practice writing x=b±b24ac2a x = \frac{-b ± \sqrt{b^2-4ac}}{2a} several times.

Why do I get two different answers?

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Quadratic equations create parabolas that can cross the x-axis at two points! Each crossing point is a solution, so x=2 x = 2 and x=12 x = -\frac{1}{2} are both correct.

Should I simplify the equation first?

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Optional but helpful! You could divide everything by 2 to get 2x23x2=0 2x^2 - 3x - 2 = 0 , making the numbers smaller and easier to work with.

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