Question

Determine whether you would use factoring, square roots, or completing the square to solve the equation. Explain your reasoning. Then solve the equation. \(2 x^2+4 x-3=0\)

Short answer

The quadratic equation \(2 x^2+4 x-3=0\) can be solved by factoring. The solutions are \(x = -1 - \sqrt{10}/2\) and \(x = -1 + \sqrt{10}/2\).

Step-by-step solution

Arranging in Standard Form

Look at the equation \(2 x^2+4 x-3=0\). This equation is already in the standard form of a quadratic equation (i.e., \(ax^2+bx+c=0\)) where \(a=2\), \(b=4\), and \(c=-3\).

Using Quadratic Formula

Using Quadratic formula, which is \(x = [-b \pm \sqrt{(b^2 - 4ac)}]/(2a)\), substitute the values of \(a\), \(b\) and \(c\), we get \(x = [-4 \pm \sqrt{(4^2 - 4(2)(-3)}]/(2*2)\). Solving the expression under square root gives \(x = [-4 \pm \sqrt{(16 + 24)}]/4\).

Solving for x

Further simplifying the above expression gives \(x = [-4 \pm \sqrt{40}]/4\). This can be simplified to \(x = [-4 \pm 2(\sqrt{10})]/4 = -1 \pm \sqrt{10}/2\). Thus, the solutions for \(x\) are \(x = -1 - \sqrt{10}/2\) and \(x = -1 + \sqrt{10}/2\).

Key concepts

Factoring Quadratic Equations

Factoring is one of the fundamental methods for solving quadratic equations. It involves rewriting the quadratic equation in a product form, where the equation is expressed as \( (x - p)(x - q) = 0 \), with \( p \) and \( q \) being the roots of the equation. This is possible when the quadratic can be expressed as the product of two binomials.

The process begins by arranging the quadratic equation in its standard form, \( ax^2 + bx + c = 0 \). The 'factoring' method works well when \( a \), \( b \) and \( c \) are integers and the equation is easily factorable. In our exercise, although the equation \( 2 x^2+4 x-3=0 \) appears suitable for factoring, sometimes the solution is not readily apparent or the numbers do not factor nicely. In this case, we might choose another method such as the quadratic formula or completing the square.

To improve understanding of factoring, one could practice recognizing patterns and common factors in different quadratic expressions, which can greatly aid in quickly identifying factorable equations.

Quadratic Formula

The quadratic formula is a powerful tool that provides the solution to any quadratic equation, regardless of whether it can be factored easily. The formula is given by \( x = [-b \pm \sqrt{b^2 - 4ac}]/(2a) \), where \( a \), \( b \) and \( c \) are the coefficients from the quadratic equation's standard form, \( ax^2 + bx + c = 0 \).

To apply the quadratic formula, one must first ensure the equation is in standard form. Then, by plugging the values from the equation into the formula, we find the values of \( x \). This method is reliable because it always works, but it involves working with square roots, which can sometimes lead to complex numbers if the discriminant (the part under the square root, \( b^2 - 4ac \)) is negative.

For our particular exercise, we used the quadratic formula to find two solutions for \( x \). It is important to remember that simplifying the square root as much as possible will lead to the simplest form of the solutions. If providing advice, giving tips on how to simplify square roots and handle negative discriminants can be beneficial to students.

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming them into a perfect square trinomial. This allows us to solve for \( x \) by taking the square root of both sides. To complete the square, one must rearrange the equation so that the \( ax^2 \) and \( bx \) terms are on one side, and the constant \( c \) is on the other. We then add and subtract the square of half the coefficient of \( x \) to the left side, to form a perfect square trinomial.

For instance, if our quadratic equation had coefficients that were more amenable, we might re-write \( 2 x^2+4 x-3=0 \) by dividing through by 2 and then adding \( 1^2 \) to each side, to construct a perfect square on the left. This technique often forms the basis of the derivation of the quadratic formula and is particularly useful when the quadratic coefficient \( a = 1 \) and the middle coefficient \( b \) is even.

As an enhancement to learning, students should be encouraged to practice completing the square with equations where \( a eq 1 \) and understand how to deal with the additional complexity this introduces.