Zeroes and Coefficients of Quadratic Polynomial

 

❓ Question

Find the zeroes of the quadratic polynomial:

$$x^2+7x+10$$

and verify the relationship between the zeroes and the coefficients.

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🖼️ Solution Image

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✍️ Short Explanation

To find the zeroes of a quadratic polynomial, we factorise it and equate each factor to zero 💯

For a quadratic polynomial:

$$ax^2+bx+c$$

if the zeroes are $\alpha$ and $\beta$, then:

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

and

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

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🔹 Step 1 — Write the Polynomial

$$f(x)=x^2+7x+10$$

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🔹 Step 2 — Factorise the Polynomial

Split the middle term:

$$x^2+7x+10$$
$$=x^2+5x+2x+10$$

Take common factors:

$$=x(x+5)+2(x+5)$$
$$=(x+2)(x+5)$$

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🔹 Step 3 — Find the Zeroes

$$x+2=0$$
$$x=-2$$

and

$$x+5=0$$
$$x=-5$$

Therefore, the zeroes are:

$$\boxed{-2 \text{ and } -5}$$

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🔹 Step 4 — Verify Relationship Between Zeroes and Coefficients

For:

$$ax^2+bx+c$$

Here,

$$a=1,\quad b=7,\quad c=10$$

Let:

$$\alpha=-2,\quad \beta=-5$$

(i) Sum of Zeroes

$$\alpha+\beta=(-2)+(-5)=-7$$

Also,

$$-\frac{b}{a}=-\frac{7}{1}=-7$$

Verified ✅

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(ii) Product of Zeroes

$$\alpha\beta=(-2)(-5)=10$$

Also,

$$\frac{c}{a}=\frac{10}{1}=10$$

Verified ✅

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✅ Final Answer

Zeroes of the polynomial are:

$$\boxed{-2 \text{ and } -5}$$

Relationship verified:

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

and

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

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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 → minus middle coefficient, Product → constant term

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📚 Related Topics

• Irrational Numbers Concept Using Contradiction Method
• Irrational Numbers Proof Using Contradiction Method
• Zeroes of Polynomial Using Graph Concept Explained