Question

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$5 x^{2}+5=20$$

Short answer

The solution to the equation are \(x = \sqrt{3}\) and \(x = -\sqrt{3}\).

Step-by-step solution

Isolate the variable

First, we isolate \(x^{2}\) on one side of the equation. This can be done by subtracting 5 from both sides of the equation, to get \(5x^{2} = 20 - 5, which simplifies to 5x^{2} = 15.

Simplify the equation

Now we simplify the equation by dividing everything by 5, to get \(x^{2} = 3\).

Solve for variable

Finally, we find the square root of both sides of the equation, remembering that square roots have both positive and negative solutions, so \(x = \sqrt{3}\) and \(x = -\sqrt{3}\).

Key concepts

Isolate Variables

When solving quadratic equations, the first key step is to isolate the variable – in other words, to get the variable on its own on one side of the equation. By doing this, we pave the way to finding the variable's value. In our exercise, the process begins by dealing with the equation 5x^2 + 5 = 20. To isolate the variable term x^2, we need to eliminate other terms from its side of the equation. This is done by subtracting 5 from both sides, leading us to 5x^2 = 15.

Afterward, to fully isolate x^2, we divide both sides by the coefficient of x^2, which is 5. This calculation gives us x^2 = 3, a much simpler expression which sets the stage for utilizing square roots to find the value of x. It's essential to handle one operation at a time and keep the equation balanced—what you do to one side must be done to the other side.

Radical Expressions

Radical expressions are mathematical expressions that include a radical symbol, which denotes the root of a number. The most common is the square root, but there can also be cube roots, fourth roots, and so on. In the context of our problem, after isolating the variable and simplifying the expression to x^2 = 3, the next step is to express the solution as a radical expression because 3 is not a perfect square, and therefore x cannot be simplified to a neat integer.

It’s crucial to understand that radical expressions can often remain in their radical form, especially when dealing with irrational numbers — those that cannot be expressed as a simple fraction. This is natural in mathematics and is a valid form of expressing the precise value of a solution. In other cases, if a solution can be simplified further because the radical is a perfect square, then it should be simplified.

Square Roots

Square roots are a particular kind of radical expression that relate to squaring. Simply put, the square root of a number is a value that, when multiplied by itself, gives the original number. The square root symbol is \( \sqrt{} \). When we derived the equation x^2 = 3, to solve for x, we had to think about what number squared equals 3.

The important concept to remember with square roots is that every positive number has two square roots: one positive and one negative. This is because both (+\sqrt{3})^2 and (-\sqrt{3})^2 will result in 3. Hence, for our original problem, there are two valid solutions: \( x = +\sqrt{3} \) and \( x = -\sqrt{3} \). This concept is essential because it ensures we account for all possible solutions to the equation.