{"id":67293,"date":"2024-02-01T09:59:00","date_gmt":"2024-02-01T09:59:00","guid":{"rendered":"https:\/\/schooltutoring.com\/help\/?p=67293"},"modified":"2024-02-06T19:58:56","modified_gmt":"2024-02-06T19:58:56","slug":"quadratic-equations","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/quadratic-equations\/","title":{"rendered":"Solving Quadratic Equations"},"content":{"rendered":"
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What are Quadratic Equations?<\/h1>\n

The general form of quadratic equations is <\/p>\n

  <\/span>   <\/span>\"\[ax^2+bx+c=0\]\"<\/p>\n

where \"x\" is our unknown \"a,b,c\" are real numbers and \"a \neq 0\". The last statement is important as if \"a\" was zero, the term \"ax^2\" would be zero and the equation wouldn’t have a quadratic term.<\/p>\n

Having a quadratic term is not enough though, you cannot have your unknown to any power bigger than \"2\" (i.e.. \"x^3\", \"x^4\", etc.) and also the expression must be a polynomial (there cannot be “weird” functions like \"\sin(x)\", \"\cos(x)\", \"\log(x)\", \"\frac{1}{x}\", \"\sqrt{x}\", \"x^{0.3}\" affecting our unknown) only non-negative integers are allowed.<\/p>\n

In practice, quadratic equations can take a lot of forms and look quite different than their general form. Let’s see some examples:<\/p>\n

  <\/span>   <\/span>\"\[3x^2+2x+3=0, \quad 2y^2-y=0, \quad \pi z^2-3=0, \quad \frac{1}{2}x^2=-3, \quad -3x-4x^2=2\]\"<\/p>\n

These are examples of equations which are not quadratic:<\/p>\n

  <\/span>   <\/span>\"\[x+2=0 \quad \text{is missing the quadratic term } x^2\]\"<\/p>\n

  <\/span>   <\/span>\"\[x^3+3x^2+x=0 \quad \text{has the unknown to a power bigger than 2} (x^3 \text{is the biggest})\]\"<\/p>\n

  <\/span>   <\/span>\"\[x^2+\frac{1}{x}=0 \quad \text{has a "weird" function affecting the unknown}\]\"<\/p>\n

  <\/span>   <\/span>\"\[x^2+x^{0.5}=0 \quad \text{has a term of the form } x^{0.5} \text{ which makes the expression not a polynomial}\]\"<\/p>\n

Now we put our attention on how to solve these equations.<\/p>\n

Solving Quadratic Equations<\/h1>\n

General Form<\/h2>\n

While solving quadratic equations we must take into consideration their form. To know the form of a quadratic equation we can put it in general form\u00a0 (G.F). General Form is no more than the equation written as \"ax^2+bx+c=0\". Notice that all the terms are on the left of the equal sign in order (\"x^2\" first and \"x\" second), and with zero on the right. Let’s see some examples of how to take a quadratic equation to G.F.<\/p>\n

  <\/span>   <\/span>\"\[3x^2=2x-3 \implies 3x^2-2x+3=0\]\"<\/p>\n

  <\/span>   <\/span>\"\[\frac{1}{2}x^2=1 \implies \frac{1}{2}x^2-1=0\]\"<\/p>\n

  <\/span>   <\/span>\"\[-3x=x^2 \implies -x^2-3x=0\]\"<\/p>\n

Solving the Quadratic Equation<\/h2>\n

The following diagram describes how to solve a quadratic equation, once it is in General Form.<\/p>\n[\/vc_column_text]

\"Diagram <\/div><\/div>[\/vc_column][\/vc_row][vc_row][vc_column][vc_column_text]\n

Factor if you can<\/h2>\n

If you thought that factoring was not useful, you are wrong. Factoring is essential to solving equations of any type, especially for quadratic equations. We can see why by remembering a key feature of numbers.<\/p>\n

“If a multiplication of numbers equals zero, then one of the factors must be zero”<\/em><\/p>\n

For example, if \"x*y*z=0\" then one of them must be zero \"x=0\", \"y=0\" or \"z=0\". Let’s see how to exploit this principle to solve equations.<\/p>\n

  <\/span>   <\/span>\"\[x^2-4x+3=0 \implies (x-3)(x-1)=0 \quad \text{By Factoring}\]\"<\/p>\n

Notice that this is a multiplication that equals zero then that means that<\/p>\n

  <\/span>   <\/span>\"\[x-3=0 \quad\text{ or }\quad x-1=0\]\"<\/p>\n

Notice that both of the resulting equations are simple linear equations! And we know how to solve them, the results are:<\/p>\n

  <\/span>   <\/span>\"\[x=3 \quad\text{ or }\quad x=1\]\"<\/p>\n

These are precisely the solutions to our equation.<\/p>\n

If you can’t factor use the Quadratic Formula<\/h2>\n

Factoring is not always easy or obvious. For those cases, or any case in fact, you always have the quadratic formula at your service. At last here it is:<\/p>\n

The solution to an equation to the form \"ax^2+bx+c=0\" is given by<\/em><\/p>\n

  <\/span>   <\/span>\"\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]\"<\/p>\n

The quadratic formula can solve any quadratic equation, but it can take a lot of steps to reach a solution. This means that you should use it only when it is necessary. We must accept the formula can look a bit scary at first. However, all you need to do is identify the values of a,b and c from your equation. Let’s solve \"x^2-4x+3=0\" using the quadratic equation.<\/span><\/p>\n

First identify \"a\",\"b\" and \"c\"<\/p>\n

  <\/span>   <\/span>\"\[\boxed{1}x^2\boxed{-4}x\boxed{+3} = 0 | \quad & \boxed{a}x^2+\boxed{b}x+\boxed{c}=0\]\"<\/p>\n

Use them in the quadratic formula<\/p>\n

  <\/span>   <\/span>\"\[x&=\frac{-(-4)\pm\sqrt{(-4)^2-4(1)(3)}}{2(1)} \quad | \quad x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]\"<\/p>\n

Simplify<\/p>\n

  <\/span>   <\/span>\"\begin{align*}<\/p>\n

In this case, we have two solutions, one given by separating \"\pm\" into plus and minus<\/p>\n

  <\/span>   <\/span>\"\begin{align*}<\/p>\n

If you notice this is the same solution we found in the previous section (As we solved the same equation). I hope you agree that by factoring we solved the equation faster.<\/p>\n

Special Cases<\/h2>\n

When the equation is missing terms, the linear term \"bx\" or the constant term \"c\". We can solve the equation easily.<\/p>\n

Missing bx<\/h3>\n

When \"b=0\" the equation looks like \"ax^2+c=0\". Examples are \"3x^2+2=0\", \"x^2-2=0\" or \"\frac{1}{2}x^2-\frac{1}{3}\".<\/p>\n

To solve this equation we need only need to isolate \"x\" (You could solve it by factoring or using the quadratic formula as well, but I think this is the easiest approach).<\/p>\n

  <\/span>   <\/span>\"\begin{align*}<\/p>\n

Missing c<\/h3>\n

When \"c=0\" the equation looks like \"ax^2+bx=0\". Examples are \"2x^2+2x=0\", \"x^2-3x=0\" or \"\frac{1}{2}x^2-\frac{1}{3}x\".<\/p>\n

To solve this equation we need only need to take the Common Factor (Factoring again!).<\/p>\n

  <\/span>   <\/span>\"\begin{align*}<\/p>\n

Notice that zero is always a solution for these types of equations.<\/p>\n

No solutions for Quadratic Equations<\/h2>\n

Sometimes, a quadratic equation has no solutions in the real numbers. This happens whenever in one of our methods we encounter the square root of a negative number (i.e. \"\sqrt{-3}, \sqrt{-4}, \sqrt{-\frac{1}{2}}\"). If you are working in real numbers and you see one of these square roots your answer should be immediately:<\/p>\n

\u00a0“There is no solution in the real numbers”<\/em><\/p>\n

As a disclaimer, these equations have solutions in complex numbers, but that is a different topic that deserves its own post.<\/p>\n

Solving Quadratic Equations Using Technology<\/h2>\n

Lastly, I want to show you how to solve a quadratic equation technology. You can use for example this widget from Wolfram Alpha.<\/p>\n[\/vc_column_text][vc_raw_html]JTNDc2NyaXB0JTIwdHlwZSUzRCUyMnRleHQlMkZqYXZhc2NyaXB0JTIyJTIwaWQlM0QlMjJXb2xmcmFtQWxwaGFTY3JpcHRiZjQyNGNiN2IwZGVhMDUwYTQyYjk3MzllYjI2MWEzYSUyMiUyMHNyYyUzRCUyMiUyRiUyRnd3dy53b2xmcmFtYWxwaGEuY29tJTJGd2lkZ2V0JTJGd2lkZ2V0LmpzcCUzRmlkJTNEYmY0MjRjYjdiMGRlYTA1MGE0MmI5NzM5ZWIyNjFhM2ElMjIlM0UlM0MlMkZzY3JpcHQlM0U=[\/vc_raw_html][\/vc_column][\/vc_row][vc_row][vc_column][vc_column_text]\n

If you need to write \"\sqrt{2}\" for example you can do so by writing “sqrt(2)” and exponents like \"3^2\" can be written as 3^2.<\/p>\n

Step-by-step solutions are available in Symbolab<\/a>. Wolfram Alpha also has this option but it is paid.<\/p>\n

Calculators and Quadratic Equations<\/h3>\n

Some calculators can solve quadratic equations as well. Check yours! You would be surprised by the power of your calculator. This video from Blackpenredpen<\/a> has an example of how to solve a quadratic equation using a Casio calculator.<\/p>\n[\/vc_column_text]