Solve the Quadratic Equation: Finding Roots in 5x² - 6x + 1

Q: Why do we use the quadratic formula instead of factoring?

The quadratic formula always works for any quadratic equation! While $ 5x^2 - 6x + 1 $ can be factored as $ (5x-1)(x-1) = 0 $, the formula is more reliable when factoring is difficult.

Q: What does the discriminant tell us?

The discriminant $ b^2 - 4ac $ reveals the nature of solutions. Since 16 > 0, we get two real solutions. If it were 0, we'd have one solution; if negative, no real solutions.

Q: How do I avoid arithmetic mistakes with the formula?

Take it step by step: First find the discriminant, then $ \sqrt{16} = 4 $, then calculate both $ \frac{6+4}{10} $ and $ \frac{6-4}{10} $ separately.

Q: Why is the answer x₁ = 1 and x₂ = 1/5, not the other way around?

The order doesn't matter mathematically! Both $ x = 1 $ and $ x = \frac{1}{5} $ are correct solutions. Some textbooks list the larger root first, others list them as they appear in calculations.

Q: Can I check my answers without substituting back?

Substitution is the most reliable way to verify! But you can also check that the solutions multiply to $ \frac{c}{a} = \frac{1}{5} $ and add to $ \frac{6}{5} $ using Vieta's formulas.

Quadratic Formula with Positive Discriminant

Solve the following equation:

$5x^2-6x+1=0$

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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

00:32 Substitute appropriate values according to the given data and solve

00:59 Calculate the square and products

01:16 Calculate the square root of 16

01:33 These are the 2 possible solutions (addition,subtraction)

01:42 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$5x^2-6x+1=0$

Step-by-step solution

To solve the quadratic equation $5x^2 - 6x + 1 = 0$ , we will use the quadratic formula:

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

Identify the coefficients: $a = 5$ , $b = -6$ , and $c = 1$ .

Step 1: Calculate the discriminant using $b^2 - 4ac$ .
The discriminant is $(-6)^2 - 4 \times 5 \times 1 = 36 - 20 = 16$ .

Step 2: Find the solutions using the quadratic formula.
$x = \frac{-(-6) \pm \sqrt{16}}{2 \times 5}$
This simplifies to $x = \frac{6 \pm 4}{10}$ .

Step 3: Calculate both solutions.
First solution: $x_1 = \frac{6 + 4}{10} = \frac{10}{10} = 1$
Second solution: $x_2 = \frac{6 - 4}{10} = \frac{2}{10} = \frac{1}{5}$ .

Therefore, the solutions to $5x^2 - 6x + 1 = 0$ are $x_1 = 1$ and $x_2 = \frac{1}{5}$ .

The correct choice from the given options is:

$x_1=1,x_2=\frac{1}{5}$

Final Answer

$x_1=1,x_2=\frac{1}{5}$

Key Points to Remember

Essential concepts to master this topic

Formula: Use $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for any quadratic
Discriminant: Calculate $(-6)^2 - 4(5)(1) = 36 - 20 = 16$ first
Check: Substitute $x = 1$ : $5(1)^2 - 6(1) + 1 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Making sign errors with the quadratic formula
Don't forget that b = -6, so -b = -(-6) = +6 in the formula! Writing -6 instead of +6 gives wrong solutions like x = -1 and x = -1/5. Always be extra careful with negative coefficients and use -b, not just b.

Practice Quiz

Test your knowledge with interactive questions

a = coefficient of x²

b = coefficient of x

c = coefficient of the constant term

What is the value of $$ c $$ in the function $$ y=-x^2+25x $$ ?

FAQ

Everything you need to know about this question

Why do we use the quadratic formula instead of factoring?

The quadratic formula always works for any quadratic equation! While $5x^2 - 6x + 1$ can be factored as $(5x-1)(x-1) = 0$ , the formula is more reliable when factoring is difficult.

What does the discriminant tell us?

The discriminant $b^2 - 4ac$ reveals the nature of solutions. Since 16 > 0, we get two real solutions. If it were 0, we'd have one solution; if negative, no real solutions.

How do I avoid arithmetic mistakes with the formula?

Take it step by step: First find the discriminant, then $\sqrt{16} = 4$ , then calculate both $\frac{6+4}{10}$ and $\frac{6-4}{10}$ separately.

Why is the answer x₁ = 1 and x₂ = 1/5, not the other way around?

The order doesn't matter mathematically! Both $x = 1$ and $x = \frac{1}{5}$ are correct solutions. Some textbooks list the larger root first, others list them as they appear in calculations.

Can I check my answers without substituting back?

Substitution is the most reliable way to verify! But you can also check that the solutions multiply to $\frac{c}{a} = \frac{1}{5}$ and add to $\frac{6}{5}$ using Vieta's formulas.

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