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There are many situations in science, engineering, economics and other fields where quadratic equations are a vital part of the mathematics used. Some real word problems can be solved using a quadratic equation. We are generally only interested in any real solutions which result. We will look at some of these applications. We employ the […]
In many cases, factorising a quadratic equation or completing the square can be long or difficult. We can instead use the quadratic formula. \( \begin{align} \displaystyle \require{AMSsymbols} \require{color}ax^2 + bx + c &= 0 \\ax^2 + bx &= -c \\x^2 + \dfrac{b}{a}x &= -\dfrac{c}{a} \\x^2 + \dfrac{b}{a}x \color{red} + \Big(\dfrac{b}{2a}\Big)^2 &= -\dfrac{c}{a} \color{red} + \Big(\dfrac{b}{2a}\Big)^2\\\Big(x+\dfrac{b}{2a}\Big)^2 […]
Not all quadratics factorise easily. For instance, $x^2+6x+2$ cannot be factorised by simple factorisation. In other words, we cannot write $x^2+6x+2$ in the form $(x-a)(x-b)$ where $a$ and $b$ are rational. An alternative way to solve this equation $x^2+6x+2$ is by completing the square. Equations of the form $ax^2+bx+c$ can be converted to the form […]
The factorised form of a quadratic equation is $(ax+b)(cx+d)=0$. Using the Null Factor law, we can solve this equation algebraically to find $x$ \( \begin{align} \displaystyle(ax+b)(cx+d) &= 0 \\ax+b &= 0 \text{ or } cx+d =0 \\x &= -\dfrac{b}{a} \text{ or } x = -\dfrac{d}{c} \end{align} \) Example 1 Solve $(x-1)(x+2)=0$. \( \begin{align} \displaystyle(x-1)(x+2) &= […]
We can find solutions to simple exponential equations where we can make equal bases and then equate exponents (indices). For example, $2^{x}=8$ can be written as $2^x = 2^3$. Therefore the solution is $x=3$. However, it is not always easy to make the same bases, such as $2^x=5$. We use $\textit{logarithms}$ to find the exact […]
We can use the laws of logarithms to write equations in different forms. This can be particularly useful if an unknown appears as an index (exponent).$$ \large 2^x=7$$For the logarithmic function, for every value of $y$, there is only one corresponding value of $x$.$$ \large y=5^x$$We can, therefore, take the logarithm of both sides of […]
The equation $a^x=y$ is an example of a general exponent equation (indicial equation), and $2^x = 32$ is an example of a more specific exponential equation (indicial equation). To solve one of these equations, it is necessary to write both sides of the equation with the same base if the unknown is an exponent (index) […]
Since the days of old, artists, as well as mathematicians, have known that there is a special, aesthetically pleasing rectangle with width $1$, length $x$, and the following property:When a square of side $1$ is removed, the remaining rectangle has the same proportions as the original rectangle. Since the new rectangle has a width $x-1$ […]
Surd Equations Reducible to Quadratic for Math Algebra is done squaring both sides for removing surds and radical expressions. Make sure to check whether the solutions are correct by substituting them into the original surd equations. Question 1 Solve \( x = \sqrt{x+2} \). \( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}x^2 &= x+2 &\color{red} \text{square both sides} […]
Trigonometric Equations Reducible to Quadratic for Math Skills are based on trigonometric identities such as;$$ \large \begin{align}\sin^2{x} + \cos^2{x} &= 1 \\1 + \cot^2{x} &= \csc^2{x} \\\tan^2{x} + 1 &= \sec^2{x} \\end{align} $$ Question 1 Solve \( 2 \cos^2{x}-3 \cos{x} + 1 = 0 \) for \( 0^\circ \le x \le 360^\circ \). \( \begin{aligned} […]