Solve the Quadratic Equation: 2x²-7x+5=0

Q: How do I know which two numbers to use when splitting the middle term?

Find two numbers that multiply to give ac (the product of the first and last coefficients) and add to give the middle coefficient. For $ 2x^2 - 7x + 5 $, you need numbers that multiply to $ 2 \times 5 = 10 $ and add to $ -7 $.

Q: Why do we get two solutions for a quadratic equation?

A quadratic equation represents a parabola that can cross the x-axis at two points. Each crossing point gives us a solution where $ y = 0 $.

Q: What if I can't factor the quadratic easily?

If factoring is difficult, you can always use the quadratic formula: $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $. This works for any quadratic equation!

Q: Can a quadratic equation have just one solution?

Yes! When the discriminant $ b^2 - 4ac = 0 $, the parabola touches the x-axis at exactly one point, giving a repeated root.

Q: What does it mean when we set each factor equal to zero?

This uses the Zero Product Property: if $ A \times B = 0 $, then either $ A = 0 $ or $ B = 0 $ (or both). So $ (2x - 5)(x - 1) = 0 $ means either $ 2x - 5 = 0 $ or $ x - 1 = 0 $.

Quadratic Equations with Factoring Method

Find the value of the parameter x.

$2x^2-7x+5=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:12 Factor into components

00:29 Take out the common factor from the parentheses

00:56 Find what makes each factor zero

01:03 Isolate the unknown, this is one solution, now let's find the second

01:18 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

Find the value of the parameter x.

$2x^2-7x+5=0$

Step-by-step solution

We will factor using trinomials, remembering that there is more than one solution for the value of X:

$2x^2-7x+5=0$

We will factor -7X into two numbers whose product is 10:

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

We will factor out a common factor:

$2x(x-1)-5(x-1)=0$

$(2x-5)(x-1)=0$

Therefore:

$x-1=0$ $x=1$

Or:

$2x-5=0$

$2x=5$

$x=2.5$

Final Answer

$x=1,x=2.5$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Break middle term using products that multiply to ac
Technique: Split $-7x$ into $-5x - 2x$ since $(-5) \times (-2) = 10$
Check: Substitute both solutions: $2(1)^2 - 7(1) + 5 = 0$ and $2(2.5)^2 - 7(2.5) + 5 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Incorrectly splitting the middle term
Don't split -7x into random pairs like -3x and -4x = wrong factoring! These numbers don't multiply to give the correct product (ac = 10). Always find two numbers that multiply to ac and add to the middle coefficient.

Practice Quiz

Test your knowledge with interactive questions

Find the value of the parameter x.

$$ 2x^2-7x+5=0 $$

FAQ

Everything you need to know about this question

How do I know which two numbers to use when splitting the middle term?

Find two numbers that multiply to give ac (the product of the first and last coefficients) and add to give the middle coefficient. For $2x^2 - 7x + 5$ , you need numbers that multiply to $2 \times 5 = 10$ and add to $-7$ .

Why do we get two solutions for a quadratic equation?

A quadratic equation represents a parabola that can cross the x-axis at two points. Each crossing point gives us a solution where $y = 0$ .

What if I can't factor the quadratic easily?

If factoring is difficult, you can always use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . This works for any quadratic equation!

How do I check if my factoring is correct?

Expand your factored form back to the original equation. For example: $(2x - 5)(x - 1) = 2x^2 - 2x - 5x + 5 = 2x^2 - 7x + 5$ ✓

Can a quadratic equation have just one solution?

Yes! When the discriminant $b^2 - 4ac = 0$ , the parabola touches the x-axis at exactly one point, giving a repeated root.

What does it mean when we set each factor equal to zero?

This uses the Zero Product Property: if $A \times B = 0$ , then either $A = 0$ or $B = 0$ (or both). So $(2x - 5)(x - 1) = 0$ means either $2x - 5 = 0$ or $x - 1 = 0$ .

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Solve the Quadratic Equation: 2x²-7x+5=0

Quadratic Equations with Factoring Method

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know which two numbers to use when splitting the middle term?

Why do we get two solutions for a quadratic equation?

What if I can't factor the quadratic easily?

How do I check if my factoring is correct?

Can a quadratic equation have just one solution?

What does it mean when we set each factor equal to zero?

🌟 Unlock Your Math Potential

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