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PSLE Foundation Mathematics

<p>Consider the quadratic function with the equation in the general form $y = ax^2 + bx + c, \ a \neq 0$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the formula, in terms of $a$, $b$ and $c$, of the solutions (roots) to the equation $ax^2 + bx + c = 0$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">It is given that the expression $b^2 - 4ac$ is called the <strong>discriminant</strong> of the equation $ax^2 + bx + c = 0$.</div><div class="list-item-paragraph">State the relationship between the number of distinct <strong>real roots</strong> of the equation $ax^2 + bx + c = 0$ and the discriminant.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">State the relationship between the number of distinct <strong>real roots</strong> of the equation $ax^2 + bx + c = 0$ and the number of points of intersections between the graph of $y = ax^2 + bx + c$ and the $x$-axis.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence form inequalities for each of the following cases, in terms of $a$, $b$ and $c$, such that</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the graph of $y = ax^2 + bx + c$ is completely above the $x$-axis,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the graph of $y = ax^2 + bx + c$ is completely below the $x$-axis,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the graph of $y = ax^2 + bx + c$ and the $x$-axis have at least one common point.</span><span class="inline-marks">[3]</span></div></li></ol></li></ol>
<p>Note that&nbsp;$a \neq 0$ throughout this question.</p>

Consider the quadratic function with the equation in the general form $y = ax^2 + bx + c, \ a \neq 0$.

Write down the formula, in terms of $a$, $b$ and $c$, of the solutions (roots) to the equation $ax^2 + bx + c = 0$.

State the relationship between the number of distinct **real roots** of the equation $ax^2 + bx + c = 0$ and the number of points of intersections between the graph of $y = ax^2 + bx + c$ and the $x$-axis.

Hence form inequalities for each of the following cases, in terms of $a$, $b$ and $c$, such that

Note that $a \neq 0$ throughout this question.

Solving Quadratic Equations Using the Quadratic Formula

Nature of Roots of Quadratic Equations

Quadratic Expressions Positive or Negative for All Real Values

IP2 Class Test (May 2025) Bonus Question

Quadratic Functions, Equations and Inequalities

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Solve the equation $x^2 - 6x - 13 = 3$ with the different methods:</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">by factorisation</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">by completing the square</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">by the quadratic formula</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Solve the equation $-4x^2 + 4x + 3 = 0$ with the different methods:</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">by factorisation</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">by completing the square</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">by the quadratic formula</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>

Solve the equation $x^2 - 6x - 13 = 3$ with the different methods:

Solve the equation $-4x^2 + 4x + 3 = 0$ with the different methods:

Factorisation of Quadratic Expressions

Solving Quadratic Equations by Completing the Square

IP2 Class Test (May 2025) Q3

<p>If $a{{x}^{2}}+bx+c=0$ , where where $a,b$ and $c$ are real numbers and $a\ne 0$, then the solutions are given by </p>
<p style="text-align: center">$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$</p>
<p>This formula for solving quadratic equations is usually used when the <strong>quadratic expression cannot be factorised easily. </strong></p>
<p style="margin-top: 1.5rem"><strong>Tip: </strong></p>
<p>Always express the quadratic equation in the general form $a{{x}^{2}}+bx+c=0$ so that the correct value for $a,b$ and $c$ can be substituted into the formula.</p>

If $a{{x}^{2}}+bx+c=0$ , where where $a,b$ and $c$ are real numbers and $a\ne 0$, then the solutions are given by 

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

This formula for solving quadratic equations is usually used when the **quadratic expression cannot be factorised easily. **

Always express the quadratic equation in the general form $a{{x}^{2}}+bx+c=0$ so that the correct value for $a,b$ and $c$ can be substituted into the formula.

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