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The quadratic formula is a powerful tool in algebra for solving equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and \(a \eq 0\). When you encounter a quadratic equation, you can always rely on this formula to find the roots.

Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), you substitute the known values of \(a\), \(b\), and \(c\) to determine the solutions for \(x\). Here are the key steps to follow:

• Identify the values \(a\), \(b\), and \(c\) within the quadratic equation.
• Plug these values into the quadratic formula.
• Simplify under the square root, known as the discriminant.
• Carry out the addition or subtraction, and division, to find the two potential solutions for \(x\).

It's important to notice that the discriminant \(b^2 - 4ac\) determines the nature of the roots:

• If it's positive, there are two real and distinct solutions.
• If it's zero, there is one real and repeated solution.
• If it's negative, as in our example, the solutions will be complex numbers.

Complex numbers come into play when the discriminant of a quadratic equation is negative, making it impossible to find real number solutions. A complex number has the form \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the imaginary unit, defined as \(\sqrt{-1}\).

In the provided solution, the discriminant is \(4 - 60\), which is negative. Therefore, the square root part of the quadratic formula \(\sqrt{b^2 - 4ac}\) is \(\sqrt{-56}\). This is where \(i\) comes in, allowing us to write the square root of a negative number. Simplifying \(\sqrt{-56}\) gives us \(7.48i\), which means that the solutions to the equation will be complex numbers. The two solutions can therefore be written as \(0.2 + 0.748i\) and \(0.2 - 0.748i\), representing points in the complex plane rather than the real number line.

Algebraic expressions are combinations of letters (variables), numbers, and operators (like +, -, *, /) that represent a particular value or set of values. In our example, \(5x^2 - 2x + 3\) is an algebraic expression that represents a parabola when plotted on a graph.

Solving an equation involving an algebraic expression like the one we have necessitates isolating the variable—for this case, \(x\). When the expression is set to zero, as in \(5x^2 - 2x + 3 = 0\), we're essentially looking for values of \(x\) where the parabola crosses the x-axis. Sadly, not all parabolas intersect the real number axis, which is why we sometimes get complex solutions.

When working with algebraic expressions, remember to:

• Perform similar operations (like combining like terms) carefully.
• Look out for opportunities to factorise expressions.
• Remember that exponents indicate the degree of the equation, which suggests how many solutions there might be.