Solving quadratic equations (OCR GCSE Maths): Revision Notes

Worked Example: Solve the quadratic equation:

• The equation is already factorised, so set each bracket equal to zero:

• Solve for $x$:

Thus, the solutions to the quadratic equation are: $x=4\quad or\quad x=−3$

Example 1: Solve the equation $x^2−3x−28=0$.

Step-by-Step Solution:

4. The equation is already set to zero, so proceed to factorising.
5. Factorise $x^2−3x−28$ into $(x−7)(x+4)=0$.
6. Set each factor equal to zero:

7. Solve for $x$:

The solutions are $x=7$ or $x=−4$.

Example 2: Solve the equation $2x^2+5x=3$.

Step-by-Step Solution:

8. First, rearrange the equation to equal zero:

9. Factorise $2x^2+5x−3$ into $(2x−1)(x+3)=0$.
10. Set each factor equal to zero:

11. Solve for $x$:

The solutions are $x=\frac{1}2$ or $x=−3$.

📑Example: Solve $5x^2−8x+12=0$ using the quadratic formula. 12. Identify $a, b,$ and $c$=

13. Substitute into the formula:

Simplifying:

14. The discriminant is negative $(−176)$, indicating no real solutions.

📑Example: Solve $x^2−4x+2=0$ using the quadratic formula.

Step 1: Identify the Coefficients

First, you need to identify the coefficients $a, b,$ and $c$ from the quadratic equation $ax^2+bx+c=0$.

For the equation:

• $a=1$ (the coefficient of $x2$)
• $b=−4$ (the coefficient of $x$)
• $c=2$(the constant term)

Step 2: Substitute into the Quadratic Formula

Now substitute these values into the quadratic formula:

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Substituting the values:

Simplify it step by step:

Step 3: Simplify and Calculate

Now, simplify the expression under the square root and calculate the two possible values of $x$.

This gives two possible values for $x$:

And:

So, the solutions are:

Step 4: Using Your Calculator Effectively

To ensure accuracy, follow these tips when using your calculator:

• Always use brackets around negative numbers to avoid errors.
• After calculating the square root, remember to compute both the plus and minus scenarios to get both possible solutions.
• Carefully enter the formula into your calculator to avoid mistakes.

📑Example: Solve $5x^2=10−3x$ using the quadratic formula.

Step 1: Rearrange the Equation

The first step is to rearrange the equation so that it equals zero. This is necessary before you can apply the quadratic formula.

Given:

Rearrange by subtracting $10$ and adding $3x$ to both sides:

Step 2: Identify the Coefficients

Identify the coefficients $a$, $b$, and $c$ from the quadratic equation $ax^2+bx+c=0$.

For the equation:

• $a=5$ (the coefficient of $x^2$)
• $b=3$ (the coefficient of $x$)
• $c=−10$ (the constant term)

Step 3: Substitute into the Quadratic Formula

Now substitute these values into the quadratic formula:

Substituting the values:

Simplify step by step:

Step 4: Calculate the Roots

Now, you will calculate the two possible values for $x$ using a calculator.

First Root:

Second Root:

Step 5: Using Your Calculator Effectively

To ensure accuracy, here are some tips for using your calculator:

• Always use brackets around negative numbers.
• Compute both the positive and negative roots to find the two possible solutions. Pressing the buttons on the calculator:

15. Start with the square root: $\sqrt{209}$
16. Use brackets for the negative sign: $-3 + \sqrt{209}$
17. Divide the result by $10$to get one root.
18. Repeat the process with $−$ instead of $+$ to find the other root.

Final Answers:

For $5x^2+3x−10=0,$ the solutions are approximately:

📑Example: Solve $x^2+10x$ by completing the square.

Step 1: Recognise the Form

First, let's rewrite the quadratic expression $x^2+10x$ in a form that can be factorised by completing the square.

The expression can be rewritten as:

Step 2: Understand the Expansion

To confirm that $(x+5)^2−25$ is indeed the same as $x^2+10x$, we expand the square:

This shows that $x^2+10$ can be expressed as $(x+5)^2−25$.

Step 3: Complete the Square

To complete the square, you must add and subtract the same value, ensuring that the equation balances. Here's how it's done:

Given:

We rewrite it as:

Now, $(x+5)^2$ is the "square", and the $−25$ "completes" it.

Step 4: Method for Completing the Square

1. If the coefficient of $x2$ is NOT 1, factor it out first. For example, if you had $2x^2+10x$, you'd factor out the $2$ first.
2. Complete the square using the formula:

In our case, $b=10$, so:

3. Solving the equation: If you need to solve for $x$, you would now isolate $x$ by moving the constant to the other side and taking the square root. Remember to consider both the positive and negative square roots!

📑Example Problem: Solve $x^2−4x=21$ by completing the square.

Step 1: Identify the coefficient of $x^2$

• In this case, the coefficient of $x^2$ is $1$, which is ideal for completing the square without needing to factor out any terms.

Step 2: Apply the Completing the Square Formula

• Rewrite the quadratic expression $x^2−4x$ using the formula for completing the square:

• Here, $b=−4$, so:

Step 3: Solve the Equation

• Substitute back into the equation:

• Add $4$ to both sides to isolate the square:

• Take the square root of both sides:

• Solve for $x$:

• Solution: The solutions are $x=7$ or $x=−3$.

📑Example Problem: Solve $4x^2−8x=21$ by completing the square.

Step 1: Factor out the Coefficient of $x2$

• The coefficient of $x2$ is $4$, so we factor it out:

Step 2: Apply the Completing the Square Formula

• Use the completing the square formula on the expression inside the brackets:

• Substitute back:

Step 3: Solve the Equation

• Expand and solve:

• Add $4$ to both sides:

• Divide both sides by $4$:

• Take the square root of both sides:

• Solve for $x$:

• Solution: The solutions are $x=3.5$ or $x=−1.5$.