Find Zeroes and Verify Relationship Between Coefficients

 

❓ Question

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients

(i) x² - 2x - 8

(ii) 4z² - 4z + 1

(iii) 6x² - 7x - 3

(iv) 4u² + 8u

(v) t² - 15

(vi) 3x² - x - 4

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🖼️ Solution Image

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✍️ Short Explanation

For any quadratic polynomial:

$$ax^2+bx+c$$

if the zeroes are:

$$\alpha,\beta$$

then:

$$\alpha+\beta=-\frac{b}{a}$$

and

$$\alpha\beta=\frac{c}{a}$$

💯

We first factorise the polynomial to find the zeroes and then verify these relationships.

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🔹 (i) $x^2-2x-8$

Factorisation

$$x^2-2x-8=(x-4)(x+2)$$

Zeroes:

$$\boxed{4,\ -2}$$

Verification

$$4+(-2)=2=-\frac{-2}{1}$$
$$4(-2)=-8=\frac{-8}{1}$$

Verified ✅

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🔹 (ii) $4x^2-4x+1$

Factorisation

$$4x^2-4x+1=(2x-1)^2$$

Zeroes:

$$\boxed{\frac12,\ \frac12}$$

Verification

$$\frac12+\frac12=1=-\frac{-4}{4}$$
$$\frac12\times\frac12=\frac14=\frac14$$

Verified ✅

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🔹 (iii) $6x^2-7x-3$

Factorisation

$$6x^2-7x-3=(3x+1)(2x-3)$$

Zeroes:

$$\boxed{-\frac13,\ \frac32}$$

Verification

$$-\frac13+\frac32=\frac76=-\frac{-7}{6}$$
$$-\frac13\times\frac32=-\frac12=\frac{-3}{6}$$

Verified ✅

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🔹 (iv) $4u^2+8u$

Factorisation

$$4u^2+8u=4u(u+2)$$

Zeroes:

$$\boxed{0,\ -2}$$

Verification

$$0+(-2)=-2=-\frac84$$
$$0\times(-2)=0=\frac04$$

Verified ✅

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🔹 (v) $t^2-15$

Factorisation

$$t^2-15=(t-\sqrt{15})(t+\sqrt{15})$$

Zeroes:

$$\boxed{\sqrt{15},\ -\sqrt{15}}$$

Verification

$$\sqrt{15}+(-\sqrt{15})=0=-\frac01$$
$$\sqrt{15}\times(-\sqrt{15})=-15=\frac{-15}{1}$$

Verified ✅

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🔹 (vi) $3x^2-x-4$

Factorisation

$$3x^2-x-4=(x+1)(3x-4)$$

Zeroes:

$$\boxed{-1,\ \frac43}$$

Verification

$$-1+\frac43=\frac13=-\frac{-1}{3}$$
$$-1\times\frac43=-\frac43=\frac{-4}{3}$$

Verified ✅

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✅ Final Answer

Polynomial	Zeroes
$x^2-2x-8$	$4,\ -2$
$4x^2-4x+1$	$\frac12,\ \frac12$
$6x^2-7x-3$	$-\frac13,\ \frac32$
$4u^2+8u$	$0,\ -2$
$t^2-15$	$\sqrt{15},\ -\sqrt{15}$
$3x^2-x-4$	$-1,\ \frac43$

All relationships are verified ✅

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⭐ Key Insight

For quadratic polynomial:

$$ax^2+bx+c$$

• Sum of zeroes:

$$-\frac{b}{a}$$

• Product of zeroes:

$$\frac{c}{a}$$

🧠 Memory Line:

Sum → opposite sign of middle term, Product → constant term over coefficient of $x^2$

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📚 Related Topics

• Number of Zeroes of a Polynomial from Graphs
• Prove that Given Numbers are Irrational
• Zeroes and Coefficients of Quadratic Polynomial