Solve the Quadratic Equation: Find X in 2x^2 - 2x = (x + 1)^2

Quadratic Equations with Standard Form Conversion

Solve the equation

2x2−2x=(x+1)2 2x^2-2x=(x+1)^2

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:07 Let's find X together.
00:11 First, we'll use some short multiplication formulas.
00:21 Next, substitute the values and expand the brackets.
00:41 Now, plug these into our equation.
01:03 Rearrange it, so one side equals zero.
01:17 Time to group the terms together.
01:23 Let's examine the coefficients.
01:30 Use the root formula here.
01:58 Substitute the values and solve for X.
02:20 Calculate the squares and multiply them.
02:36 Factor twenty into four times five.
02:41 Break down the root of each factor.
02:47 Find the square root of four.
02:57 These are your two solutions: one by addition, one by subtraction.
03:30 And that's how we solve the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Solve the equation

2x2−2x=(x+1)2 2x^2-2x=(x+1)^2

2

Step-by-step solution

The given equation is:

2x2−2x=(x+1)2 2x^2 - 2x = (x+1)^2

Step 1: Expand the right-hand side.

(x+1)2=x2+2x+1(x+1)^2 = x^2 + 2x + 1

Step 2: Write the full equation with the expanded form.

2x2−2x=x2+2x+12x^2 - 2x = x^2 + 2x + 1

Step 3: Bring all terms to one side of the equation to set it to zero.

2x2−2x−x2−2x−1=02x^2 - 2x - x^2 - 2x - 1 = 0

Step 4: Simplify the equation.

x2−4x−1=0x^2 - 4x - 1 = 0

Step 5: Identify coefficients for the quadratic formula.

Here, a=1a = 1, b=−4b = -4, c=−1c = -1.

Step 6: Apply the quadratic formula.

x=−(−4)±(−4)2−4⋅1⋅(−1)2⋅1x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}

x=4±16+42x = \frac{4 \pm \sqrt{16 + 4}}{2}

x=4±202x = \frac{4 \pm \sqrt{20}}{2}

x=4±252x = \frac{4 \pm 2\sqrt{5}}{2}

x=2±5x = 2 \pm \sqrt{5}

Therefore, the solutions are x=2+5x = 2 + \sqrt{5} and x=2−5x = 2 - \sqrt{5}.

These solutions correspond to choice (4): Answers a + b

3

Final Answer

Answers a + b

Key Points to Remember

Essential concepts to master this topic
  • Expand: Always expand binomial squares before moving terms to one side
  • Technique: Use quadratic formula when x2−4x−1=0 x^2 - 4x - 1 = 0 doesn't factor easily
  • Check: Substitute x=2+5 x = 2 + \sqrt{5} back: both sides equal 14+45 14 + 4\sqrt{5} ✓

Common Mistakes

Avoid these frequent errors
  • Moving terms without expanding the right side first
    Don't move terms from 2x2−2x=(x+1)2 2x^2 - 2x = (x+1)^2 without expanding first = you'll miss the x2+2x+1 x^2 + 2x + 1 terms! This leads to completely wrong coefficients. Always expand binomial squares before rearranging terms to standard form.

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 can't I just take the square root of both sides?

+

You can only take square roots when you have something like x2=9 x^2 = 9 . Here, the left side 2x2−2x 2x^2 - 2x isn't a perfect square, so you need to rearrange to standard form first.

How do I know when to use the quadratic formula?

+

Use the quadratic formula when your equation is in the form ax2+bx+c=0 ax^2 + bx + c = 0 and it doesn't factor easily. If you can't find two numbers that multiply to give ac and add to give b, use the formula!

Why do I get two different answers?

+

Quadratic equations naturally have two solutions because a parabola crosses the x-axis at two points. Both x=2+5 x = 2 + \sqrt{5} and x=2−5 x = 2 - \sqrt{5} are correct!

What does the discriminant tell me?

+

The discriminant is the part under the square root: b2−4ac b^2 - 4ac . Here it's 20, which is positive, so you get two real solutions. If it were negative, you'd have no real solutions.

Can I simplify the square root further?

+

Yes! 20=4⋅5=25 \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} . Always look for perfect square factors to simplify radicals in your final answer.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Short Multiplication Formulas questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

More Questions

Click on any question to see the complete solution with step-by-step explanations

Square of sum

Area of Square with Side Length 5+2x: Express in Algebraic Form
Click to solve
Calculate the Product: Solve (x+y)² = 1 and (x²+y²)/(x+y)²=3
Click to solve

The Quadratic Formula

Solve X²+10X+9=0: Complete Quadratic Equation Guide
Click to solve
Solve Quadratic Equation: -2X² + 6X + 8 = 0
Click to solve

Combining Short Multiplication Formulas

Solve the Quadratic Equation: Finding X in 7x + 1 + (2x + 3)² = (4x + 2)²
Click to solve