Algebra

2x^2 - 7x + 3 = 0

Step-by-step solution with explanation

Final Answer

and

Step-by-step solution

1

Write down the quadratic equation

We start by identifying the equation we need to solve. It is a quadratic (highest power is 2), so it can have up to two solutions.
2

Identify the coefficients a, b, c

In any quadratic , we label the three numbers. Here , , and . We need these for the quadratic formula.
3

Plug values into the quadratic formula

The quadratic formula always works for equations in the form . We substitute our values of , , and directly into the formula.
4

Simplify the discriminant under the square root

We compute and . Subtracting gives . Since this is positive, we will get two real solutions.
5

Compute both solutions using ±

, so the formula becomes . The means we split into two separate calculations — one with and one with .
6

Solve for both values of x

Adding gives and subtracting gives . These are our two solutions.
7

Verify both solutions in the original equation

Plugging each answer back into the original equation confirms both work. Always check your answers to catch any arithmetic errors.

Understanding this problem

Learning Insight

The discriminant () is the key to a quadratic. When it equals a perfect square (like 25), the solutions are nice rational numbers. This happens because the original equation can also be factored — . The quadratic formula and factoring always give the same answers.

Quick Tip

Before using the quadratic formula, quickly check if the discriminant is a perfect square. If it is, the equation factors cleanly and you can try factoring first: find two numbers that multiply to and add to . Those numbers are and , leading straight to .

Common Mistake

Students often forget to apply the negative sign to in . Here , so . Writing as instead of leads to completely wrong answers.