Problem 63 - Algebra 1

Question

Use the quadratic formula to solve the equation. $$2 x^{2}-6 x+4=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 1\) and \(x = 2\).

Step 1: Identify the Coefficients

In the quadratic equation \(2 x^{2}-6 x+4=0\), \(a\), \(b\), and \(c\) represent the coefficients as follows: \(a = 2\), \(b = -6\), and \(c = 4\).

Step 2: Substitute into the Quadratic Formula

Now that we know the values of \(a\), \(b\), and \(c\), substitute these into the quadratic formula \(-\frac{b \pm \sqrt{b^{2}-4 a c}}{2 a}\). When you do that, you get \(-\frac{-6 \pm \sqrt{(-6)^{2}-4 \cdot 2 \cdot 4}}{2 \cdot 2}\).

Step 3: Simplify Under The Square Root

Start by simplifying the expression under the square root sign. \((-6)^{2}-4 \cdot 2 \cdot 4 = 36 - 32 = 4.\) The expression now becomes \(-\frac{-6 \pm \sqrt{4}}{2 \cdot 2}\).

Step 4: Simplify Further

The square root of 4 is 2, so the equation simplifies to \(-\frac{-6 \pm 2}{4}\). By applying the plus-minus, we get two possible solutions: \(-\frac{-6 + 2}{4}\) and \(-\frac{-6 - 2}{4}\). This further simplifies to \(-\frac{-4}{4}\) and \(-\frac{-8}{4}\).

Step 5: Final Answer

Finally, simplifying \(-\frac{-4}{4}\) yields 1 and simplifying \(-\frac{-8}{4}\) yields 2. So, the solutions to the equation \(2x^2 - 6x + 4 = 0\) are \(x = 1\) and \(x = 2\).

Key Concepts

Solving Quadratic Equations

Quadratic equations can be solved using various methods, but one effective way is by employing the quadratic formula. For any equation in the form \(ax^2 + bx + c = 0\), the solutions can be found using the formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] The symbol \(\pm\) indicates that there will typically be two solutions. This dual nature is a hallmark of solving quadratic equations.
It allows us to find possible values for \(x\) that make the entire equation equal zero. Always start by identifying the coefficients \(a\), \(b\), and \(c\) before moving on to substitution and computation.

Coefficients

Coefficients are the numbers in front of the terms in an equation. In the quadratic equation \(2x^2 - 6x + 4 = 0\), the coefficients are crucial for applying the quadratic formula correctly. Here, we have:

• \(a = 2\)
• \(b = -6\)
• \(c = 4\)

Understanding coefficients is essential because they directly influence the solutions. The values of \(a\), \(b\), and \(c\) are substituted into the quadratic formula to find the roots or solutions of the equation.
Make sure not to misplace these values as it will affect the entire calculation.

Square Root

A square root symbolizes a value that, when multiplied by itself, results in the original number. In the quadratic formula, the expression under the square root sign is \(b^2 - 4ac\) and is called the discriminant. This part is important because it determines the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is one real root, or the roots are repeated.
• If it is negative, there are no real roots, only complex roots.

In the example, simplifying \((-6)^2 - 4 \cdot 2 \cdot 4\) gives \(4\), showing a positive discriminant and hence two real roots. Square roots must be handled carefully to ensure accuracy in the final result.

Simplifying Expressions

Simplifying expressions is an important part of solving quadratic equations as it involves reducing equations to their simplest form for easier computation. After substituting values into the quadratic formula, the next step is to simplify any complex expression. In this scenario, simplify the expression \((-6)^2 - 4\cdot2\cdot4\) to get \(4\), and then take \(\sqrt{4} = 2\). This leads to further simplifications:

• Calculate both possibilities for \(-\frac{-6 \pm 2}{4}\):
• For \(+\), simplification yields \(1\).
• For \(-\), simplification yields \(2\).

This final step renders the solution complete and ensures the most accurate results.